Problems & Puzzles: Puzzles

Puzzle 1077 These numbers that are... Zak Seidov sent the following nice puzzle Some numbers that are  sum of m=3 consecutive primes and also product of 3 consecutive primes: {33263, 7566179, 10681031, 29884301}. Example: 33263 = 11083 + 11087 + 11093 = 29*31*37. Q. Find more cases with m=5, 7, 9, ...

---

During the week ending on March, contributions came from Adam Stinchcombe, Giorgos Kalogeropoulos, Emmanuel Vantieghem, Gennady Gusev, Jean-Marc REBERT, Oscar Volpatti.

***

Adam wrote:

The product of the 13 consecutive primes from 18121 to 18223 is the sum of the consecutive primes from 1808428663367515289053240288029870491511801302236142987 to 1808428663367515289053240288029870491511801302236144787.  Also, found a solution for 23 starting (for the product) at 24007 and, for the sum, starting at

2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490138629.  I think you can probably solve this for any (odd) count of factors/summands.
 

***

Giorgos wrote:

The biggest m that I found is m=39

39 -> 1359901080697656101101269858943375762235853464604777342142069921046359853043475482185086005153
4403088868540790314581931248328212636247 (134 digits)

39 consecutive prime factors for product: {2423, 2437, 2441, .......2719, 2729}

39 consecutive prime factors for sum:
 

{348692584794270795154171758703429682624577811437122395421043569499066628985506533893611796193189822
791501045905502100801239184932209,.......,34869258479427079515417175870342968262457781143712239542104356
9499066628985506533893611796193189822791501045905502100801239184945209}
 

The complete list of examples for m={3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39} with all the prime factors is in the puzzle1077.txt file that I'm sending you.

***

Emmanuel wrote:

Here are some numbers which I call "double decomposable numbers" :

m = 5 : 2775683761181, 10945513774549181, 31285407706348267, 43861128120750079
m = 7 : 52139749485151463, 988306587072911307211, 423323652722432566471943, 751633923866053302211099
m = 9 : 31359251876786281892441299570699, 706717088340549045946649789170625171, 2903138056037484201
7103801914781924933
m = 11 : 2385018819218440287149, 34237170618740076749345616844758544786453
m = 13 : 23509572623777698757692123744388316389653416929069870587
m = 15 : 436178570920976645136650311902311012102337977560516289614008518576769313
m = 17 : 166345108784858794943225366868487068031523855419640057875257310044811
m = 19 : 40522436742464107475016259525657164419018295812013777884155793803538613819423269
m = 21 : 23627359084542017700443765883390304876849497211964402702757103187986538276000593
m = 23 : 60774447238043444828454541444960992901486086159678662537487565930308633350089334028209762
683273263623
m = 25 : 74083866419045279696656253773086107876541342851489261866571473370710258057643496656942142
5040700787269
m = 27 : 12780274072995637705351351322593908728841611166146035767498453457932330552191436057459407
845862111339342001301009074914115233
m = 29 : 26855904125483404307436384151858179310346855255792339142267117777017572820925336894252360
5789802892354099485010025542709791589
m = 31 : 47019897751227719637202941561501355312813215977394462466807139510973260407015125000998092
8223630548938280265211094006690934608228604695039387583496375256829
m = 33 : 20045376743972442029343220344455261708845560709315261212198437710449636285530817336703165
776707484680133351589522293922682213734750649719845227978996083
m = 35 : 37443268978363428706702575021454603001662311425561744965277029908030346440792325123021306
6430482437655846383539477570509752080334475335121839799250447545505542772111353
m = 37 : 88955193002856011455729819991106677798813529431385604993983710755850352954529943861793115
1465648049971313606876407994828822398626415153250755662034894542847529947
m = 39 : 13599010806976561011012698589433757622358534646047773421420699210463598530434754821850860
051534403088868540790314581931248328212636247
m = 41 : 47073706617799335789398133545841990275447648434755370974234483091760045828077257709452657256590
410854307285103836639979493365287539942469070246725994198712033691189875458166883442482284230078608723
6367750538162206183648889111
m = 43 : 339059119875716093104136104836811008916447191332135813408234213901509764534797776961062146106398808
33148897592832837933202575101653030200867770383274076928448747901822638779855661662713

As can be seen, they become very big with growing  m.
Therefore we might enter a region of probable primes instead of proved primes.
 

Nevertheless I think that there are solutions for every  m. See the following .txt file for more details.

_***

Gennady wrote:

I found solutions for every odd m from 5 to 25:

 

My notation (for m=3):

33263, [29*31*37], [11083, 11087, 11093] means 33263= 29*31*37 = 11083 + 11087 + 11093 .

 

m=5

 

2775683761181, [293, 307, 311, 313, 317], [555136752211, 555136752221, 555136752227, 555136752251, 555136752271]

10945513774549181, [1607, 1609, 1613, 1619, 1621], [2189102754909767, 2189102754909797, 2189102754909853, 2189102754909881, 2189102754909883]

 

31285407706348267, [1979, 1987, 1993, 1997, 1999], [6257081541269593, 6257081541269599, 6257081541269651, 6257081541269707, 6257081541269717]

 

m=7

52139749485151463, [229, 233, 239, 241, 251, 257, 263], [7448535640735789, 7448535640735843, 7448535640735867, 7448535640735877, 7448535640735991, 7448535640736009, 7448535640736087]

988306587072911307211, [977, 983, 991, 997, 1009, 1013, 1019], [141186655296130186657, 141186655296130186669, 141186655296130186727, 141186655296130186733, 141186655296130186759, 141186655296130186823, 141186655296130186843]

423323652722432566471943, [2351, 2357, 2371, 2377, 2381, 2383, 2389], [60474807531776080924421, 60474807531776080924457, 60474807531776080924487, 60474807531776080924517,    60474807531776080924553, 60474807531776080924727, 60474807531776080924781]

 

m=9

31359251876786281892441299570699,

 [3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191],

[ 3484361319642920210271255507593, 3484361319642920210271255507619, 3484361319642920210271255507719, 3484361319642920210271255507767,  3484361319642920210271255507923, 3484361319642920210271255507937, 3484361319642920210271255507941, 3484361319642920210271255508067, 3484361319642920210271255508133]

 

706717088340549045946649789170625171,

 [9587, 9601, 9613, 9619, 9623, 9629, 9631, 9643, 9649],

[78524120926727671771849976574513799, 78524120926727671771849976574513847,

  78524120926727671771849976574513857, 78524120926727671771849976574513869,

  78524120926727671771849976574513899, 78524120926727671771849976574513911,

  78524120926727671771849976574513953, 78524120926727671771849976574514003,

  78524120926727671771849976574514033]

 

m=11

2385018819218440287149,

 [67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109], 

 [216819892656221844131, 216819892656221844133,

 216819892656221844139, 216819892656221844169,

 216819892656221844307, 216819892656221844331,

 216819892656221844347, 216819892656221844373,

 216819892656221844397, 216819892656221844401,

 216819892656221844421]

 

34237170618740076749345616844758544786453,

 [4793, 4799, 4801, 4813, 4817, 4831, 4861, 4871, 4877, 4889, 4903], 

 [3112470056249097886304146985887140434683,

 3112470056249097886304146985887140434833,

 3112470056249097886304146985887140434873,

 3112470056249097886304146985887140434881,

 3112470056249097886304146985887140435061,

 3112470056249097886304146985887140435091,

 3112470056249097886304146985887140435113,

 3112470056249097886304146985887140435197,

 3112470056249097886304146985887140435481,

 3112470056249097886304146985887140435611,

 3112470056249097886304146985887140435629]

 

m=13

23509572623777698757692123744388316389653416929069870587,

[18121, 18127, 18131, 18133, 18143, 18149, 18169, 18181, 18191, 18199, 18211, 18217, 18223], [1808428663367515289053240288029870491511801302236142987, 1808428663367515289053240288029870491511801302236142999, 1808428663367515289053240288029870491511801302236143013, 1808428663367515289053240288029870491511801302236143293, 1808428663367515289053240288029870491511801302236143791, 1808428663367515289053240288029870491511801302236143839, 1808428663367515289053240288029870491511801302236143997, 1808428663367515289053240288029870491511801302236144109, 1808428663367515289053240288029870491511801302236144279, 1808428663367515289053240288029870491511801302236144429, 1808428663367515289053240288029870491511801302236144511, 1808428663367515289053240288029870491511801302236144553, 1808428663367515289053240288029870491511801302236144787]

 

m=15

 

436178570920976645136650311902311012102337977560516289614008518576769313, 

[59629, 59651, 59659, 59663, 59669, 59671, 59693, 59699, 59707, 59723, 59729, 59743, 59747, 59753, 59771],

    [29078571394731776342443354126820734140155865170701085974267234571782657,

    29078571394731776342443354126820734140155865170701085974267234571782997, 

    29078571394731776342443354126820734140155865170701085974267234571783227, 

    29078571394731776342443354126820734140155865170701085974267234571783891, 

    29078571394731776342443354126820734140155865170701085974267234571784103, 

    29078571394731776342443354126820734140155865170701085974267234571784143, 

    29078571394731776342443354126820734140155865170701085974267234571784659, 

    29078571394731776342443354126820734140155865170701085974267234571785009, 

    29078571394731776342443354126820734140155865170701085974267234571785061, 

    29078571394731776342443354126820734140155865170701085974267234571785301, 

    29078571394731776342443354126820734140155865170701085974267234571785331,

    29078571394731776342443354126820734140155865170701085974267234571785477, 

    29078571394731776342443354126820734140155865170701085974267234571785763, 

    29078571394731776342443354126820734140155865170701085974267234571785841,

    29078571394731776342443354126820734140155865170701085974267234571785853]

 

901080977455458361174593070521399650681704862478859065575973244623246469, 

[62591, 62597, 62603, 62617, 62627, 62633, 62639, 62653, 62659, 62683, 62687, 62701, 62723, 62731, 62743], 

[60072065163697224078306204701426643378780324165257271038398216308215353, 

60072065163697224078306204701426643378780324165257271038398216308215479, 

60072065163697224078306204701426643378780324165257271038398216308215779, 

60072065163697224078306204701426643378780324165257271038398216308215941, 

60072065163697224078306204701426643378780324165257271038398216308215953, 

60072065163697224078306204701426643378780324165257271038398216308216253, 

60072065163697224078306204701426643378780324165257271038398216308216269, 

60072065163697224078306204701426643378780324165257271038398216308216431, 

60072065163697224078306204701426643378780324165257271038398216308216607, 

60072065163697224078306204701426643378780324165257271038398216308216623, 

60072065163697224078306204701426643378780324165257271038398216308216907, 

60072065163697224078306204701426643378780324165257271038398216308216911,

60072065163697224078306204701426643378780324165257271038398216308216973,

60072065163697224078306204701426643378780324165257271038398216308217471, 

60072065163697224078306204701426643378780324165257271038398216308217519]

 

m=17

166345108784858794943225366868487068031523855419640057875257310044811, 

[10247, 10253, 10259, 10267, 10271, 10273, 10289, 10301, 10303, 10313, 10321, 10331, 10333, 10337, 10343, 10357, 10369], [9785006399109340879013256874616886354795520907037650463250430001893,

 9785006399109340879013256874616886354795520907037650463250430001959,

 9785006399109340879013256874616886354795520907037650463250430002063,

 9785006399109340879013256874616886354795520907037650463250430002069,

 9785006399109340879013256874616886354795520907037650463250430002139,

 9785006399109340879013256874616886354795520907037650463250430002189,

 9785006399109340879013256874616886354795520907037650463250430002369,

 9785006399109340879013256874616886354795520907037650463250430002379,

 9785006399109340879013256874616886354795520907037650463250430002469,

 9785006399109340879013256874616886354795520907037650463250430002481,

 9785006399109340879013256874616886354795520907037650463250430002783,

 9785006399109340879013256874616886354795520907037650463250430002943,

 9785006399109340879013256874616886354795520907037650463250430003297,

 9785006399109340879013256874616886354795520907037650463250430003311,

 9785006399109340879013256874616886354795520907037650463250430003401,

 9785006399109340879013256874616886354795520907037650463250430003503,

 9785006399109340879013256874616886354795520907037650463250430003563]

 

 m=19

 40522436742464107475016259525657164419018295812013777884155793803538613819423269, 

 [15391, 15401, 15413, 15427, 15439, 15443, 15451, 15461, 15467, 15473, 15493, 15497, 15511, 15527, 15541, 15551, 15559, 15569, 15581], [2132759828550742498685066290824061285211489253263883046534515463344137569441423, 2132759828550742498685066290824061285211489253263883046534515463344137569441643, 2132759828550742498685066290824061285211489253263883046534515463344137569441937, 2132759828550742498685066290824061285211489253263883046534515463344137569442329, 2132759828550742498685066290824061285211489253263883046534515463344137569442849, 2132759828550742498685066290824061285211489253263883046534515463344137569442881, 2132759828550742498685066290824061285211489253263883046534515463344137569442993, 2132759828550742498685066290824061285211489253263883046534515463344137569443017, 2132759828550742498685066290824061285211489253263883046534515463344137569443121, 2132759828550742498685066290824061285211489253263883046534515463344137569443313, 2132759828550742498685066290824061285211489253263883046534515463344137569443443, 2132759828550742498685066290824061285211489253263883046534515463344137569443779, 2132759828550742498685066290824061285211489253263883046534515463344137569443853, 2132759828550742498685066290824061285211489253263883046534515463344137569443887, 2132759828550742498685066290824061285211489253263883046534515463344137569444091, 2132759828550742498685066290824061285211489253263883046534515463344137569444477, 2132759828550742498685066290824061285211489253263883046534515463344137569444693, 2132759828550742498685066290824061285211489253263883046534515463344137569444703, 2132759828550742498685066290824061285211489253263883046534515463344137569444837]

 

m=21

23627359084542017700443765883390304876849497211964402702757103187986538276000593,

 [5903, 5923, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113], [1125112337359143700021131708732871660802357010093542985845576342285073251235473, 1125112337359143700021131708732871660802357010093542985845576342285073251235559, 1125112337359143700021131708732871660802357010093542985845576342285073251235733, 1125112337359143700021131708732871660802357010093542985845576342285073251235853, 1125112337359143700021131708732871660802357010093542985845576342285073251235983, 1125112337359143700021131708732871660802357010093542985845576342285073251236309, 1125112337359143700021131708732871660802357010093542985845576342285073251236757, 1125112337359143700021131708732871660802357010093542985845576342285073251236921, 1125112337359143700021131708732871660802357010093542985845576342285073251237251, 1125112337359143700021131708732871660802357010093542985845576342285073251238223, 1125112337359143700021131708732871660802357010093542985845576342285073251238407, 1125112337359143700021131708732871660802357010093542985845576342285073251238857, 1125112337359143700021131708732871660802357010093542985845576342285073251238919, 1125112337359143700021131708732871660802357010093542985845576342285073251239571, 1125112337359143700021131708732871660802357010093542985845576342285073251239769, 1125112337359143700021131708732871660802357010093542985845576342285073251239771, 1125112337359143700021131708732871660802357010093542985845576342285073251239891, 1125112337359143700021131708732871660802357010093542985845576342285073251240039, 1125112337359143700021131708732871660802357010093542985845576342285073251240219, 1125112337359143700021131708732871660802357010093542985845576342285073251240339, 1125112337359143700021131708732871660802357010093542985845576342285073251240749]

 

m=23

60774447238043444828454541444960992901486086159678662537487565930308633350089334028209762683273263623,

[24007, 24019, 24023, 24029, 24043, 24049, 24061, 24071, 24077, 24083, 24091, 24097, 24103, 24107, 24109, 24113, 24121, 24133, 24137, 24151, 24169, 24179, 24181], [2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490138629, 2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490139061, 2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490139383, 2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490139397, 2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490139581, 2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490139607, 2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490139901, 2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490140747, 2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490140991, 2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490141207, 2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490141291, 2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490141293, 2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490142371, 2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490142553, 2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490143141, 2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490143387, 2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490143499, 2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490143687, 2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490144237, 2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490144239, 2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490144821, 2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490145023, 2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490145577]

 

m=25

740838664190452796966562537730861078765413428514892618665714733707102580576434966569421425040700787269,

[11783, 11789, 11801, 11807, 11813, 11821, 11827, 11831, 11833, 11839, 11863, 11867, 11887, 11897, 11903, 11909, 11923, 11927, 11933, 11939, 11941, 11953, 11959, 11969, 11971], [29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628028993, 29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628029083, 29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628029227, 29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628029343, 29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628029491, 29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628029773, 29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628030313, 29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628030613, 29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628030877, 29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628031149, 29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628031311, 29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628031471, 29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628031591, 29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628031941, 29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628032037, 29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628032121, 29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628032163, 29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628032409, 29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628032683, 29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628032961, 29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628033243, 29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628033417, 29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628033573, 29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628033673, 29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628033813]

***

Jean-Marc wrote:

More cases with m <= 49.

 

m,  n,  first prime of sum, first prime of product

1, 2, 2, 2

3, 33263, 11083, 29

5, 2775683761181, 555136752211, 293

7, 52139749485151463, 7448535640735789, 229

9, 31359251876786281892441299570699, 3484361319642920210271255507593, 3119

11, 2385018819218440287149, 216819892656221844131, 67

13, 23509572623777698757692123744388316389653416929069870587, 1808428663367515289053240288029870491511801302236142987, 18121

15, 436178570920976645136650311902311012102337977560516289614008518576769313, 290785713947317763424433541268207341401558651
70701085974267234571782657, 59629

17, 166345108784858794943225366868487068031523855419640057875257310044811, 978500639910934087901325687461688635479552090703
7650463250430001893, 10247

19, 40522436742464107475016259525657164419018295812013777884155793803538613819423269, 2132759828550742498685066290824061285
211489253263883046534515463344137569441423, 15391

21, 23627359084542017700443765883390304876849497211964402702757103187986538276000593, 1125112337359143700021131708732871660
802357010093542985845576342285073251235473, 5903

23, 60774447238043444828454541444960992901486086159678662537487565930308633350089334028209762683273263623, 2642367271219280
209932806149780912734847221137377333153803807214361244928264753653400424464490138629, 24007

25, 740838664190452796966562537730861078765413428514892618665714733707102580576434966569421425040700787269, 296335465676181
11878662501509234443150616537140595704746628589348284103223057398662776857001628028993, 11783

27, 12780274072995637705351351322593908728841611166146035767498453457932330552191436057459407845862111339342001301009074914
115233, 4733434841850236187167167156516262492163559691165198432406834614049011315626457799059039942911893088645185667040398
11630439, 39359

29, 26855904125483404307436384151858179310346855255792339142267117777017572820925336894252360578980289235409948501002554270
9791589, 926065659499427734739185660408902734839546732958356522147141992310950786928459892905253813068285835703791327620777
7334816371, 21013

31, 4701989775122771963720294156150135531281321597739446246680713951097326040701512500099809282236305489382802652110940066
90934608228604695039387583496375256829, 1516770895200894181845256179403269526219781160561111692477649661644298722806939516
1612288007213888675428395651970774409384987362213054678689922048270164507, 104917

33, 2004537674397244202934322034445526170884556070931526121219843771044963628553081733670316577670748468013335158952229392
2682213734750649719845227978996083, 60743565890825581907100667710470490026804729422167458218783144577120109956153991929403
5326566893475155556108773402846141885264689413627874097817540833, 38273

35, 3744326897836342870670257502145460300166231142556174496527702990803034644079232512302130664304824376558463835394775705
09752080334475335121839799250447545505542772111353, 1069807685096097963048645000612988657190351755016049856150772283086581
3268797807178006087612299498218738468101127930585992916580985009574909708550012787014444079197177, 61129

37, 8895519300285601145572981999110667779881352943138560499398371075585035295452994386179311514656480499713136068764079948
28822398626415153250755662034894542847529947, 2404194405482594904208914053813693994562527822469881216053613804212171701473
7822665349490580152649999224692077740756616995199962876085222993396271213366022899613, 23663

39, 135990108069765610110126985894337576223585346460477734214206992104635985304347548218508600515344030888685407903145819
31248328212636247, 348692584794270795154171758703429682624577811437122395421043569499066628985506533893611796193189822791
501045905502100801239184932209, 2423

41, 470737066177993357893981335458419902754476484347553709742344830917600458280772577094526572565904108543072851038366399
794933652875399424690702467259941987120336911898754581668834424822842300786087236367750538162206183648889111, 11481391857
9998379974141789136199976281579630328671636522523129492097672751407945632811359162415636230017768545943024340
22772021351205480261035786827853344398461265823282479727668898118104897221639911408549711273321552401973, 301627

43, 339059119875716093104136104836811008916447191332135813408234213901509764534797776961062146106398808331488975928328379
33202575101653030200867770383274076928448747901822638779855661662713, 788509581106316495591014197294909323061505096121246
077693567939305836661708832039444330572340462344956951106810065998446571513991930934903901636820327370429040648879596250
694317702783, 19391

45, 246860999375281883059434470268017926067817720952411371104576594944615010052930612900261090887601157247367030546370272
827857433688891906527780618306532527447529885630234254330906079663281018780616635210048989621501275192947750406644471, 548579998611737517909854378373373169039594935449803046899059099876922244562068028667246868639113682771926734547489495173
0165193086486811728458184589611721056219680671872318464579548072911528458147449112199769366695004287727786804869, 153379

47, 13971014839499231494812140385850431643470595274361976012118276768314729470099962684690755179837441233063533963156684
59914157470525724737863045937377746015195294109, 297255634882962372230045540124477269010012665411956936428048441879036797236169418823207557017817898575819871556525204237
05478096292015699213743348462681174356343, 2663

49, 17236912803945807396198763739703723081969077917906125331189260634945404421706550620579418117158267573363561639542107302
09876462282692759347104661676343672182093630156201105647315711790284545388138159662247781, 3517737306927715795142604844837494506524301615899209251263114415294980494225826657261105738195564810890522783580021898
3875029842503933864226625748496809636369257758289818482598279832454786640574248156361281, 16699

***

Oscar wrote:

For 5 <= m <= 29,  I found the smallest integer x which is both the product of m consecutive primes p_k and the sum of m consecutive primes q_k.

Addends q_k are about as large as x/m,  so I saved the integer offsets r_k = m*q_k - x,  much smaller (usually not larger than log(x)*m^2).

 

m = 3;

smallest x:

33263;

factors:

29, 31, 37;

offsets:

-14, -2, 16.

 

m = 5;

smallest x:

2775683761181;

factors:

293, 307, 311, 313, 317;

offsets:

-126, -76, -46, 74, 174.

 

m = 7;

smallest x:

52139749485151463;

factors:

229, 233, 239, 241, 251, 257, 263;

offsets:

-940, -562, -394, -324, 474, 600, 1146.

 

m = 9;

smallest x:

31359251876786281892441299570699;

factors:

3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191;

offsets:

-2362, -2128, -1228, -796, 608, 734, 770, 1904, 2498.

 

m = 11;

smallest x:

2385018819218440287149;

factors:

67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109;

offsets:

-1708, -1686, -1620, -1290, 228, 492, 668, 954, 1218, 1262, 1482.

 

m = 13;

smallest x:

23509572623777698757692123744388316389653416929069870587;

factors:

18121, 18127, 18131, 18133, 18143, 18149, 18169, 18181, 18191, 18199, 18211, 18217, 18223;

offsets:

-11756, -11600, -11418, -7778, -1304, -680, 1374, 2830, 5040, 6990, 8056, 8602, 11644.

 

m = 15;

smallest x:

436178570920976645136650311902311012102337977560516289614008518576769313;

factors:

59629, 59651, 59659, 59663, 59669, 59671, 59693, 59699, 59707, 59723, 59729, 59743, 59747, 59753, 59771;

offsets:

-29458, -24358, -20908, -10948, -7768, -7168, 572, 5822, 6602, 10202, 10652, 12842, 17132, 18302, 18482.

 

m = 17;

smallest x:

166345108784858794943225366868487068031523855419640057875257310044811;

factors:

10247, 10253, 10259, 10267, 10271, 10273, 10289, 10301, 10303, 10313, 10321, 10331, 10333, 10337, 10343, 10357, 10369;

offsets:

-12630, -11508, -9740, -9638, -8448, -7598, -4538, -4368, -2838, -2634, 2500, 5220, 11238, 11476, 13006, 14740, 15760.

 

m = 19;

smallest x:

40522436742464107475016259525657164419018295812013777884155793803538613819423269;

factors:

15391, 15401, 15413, 15427, 15439, 15443, 15451, 15461, 15467, 15473, 15493, 15497, 15511, 15527, 15541, 15551, 15559, 15569, 15581;

offsets:

-36232, -32052, -26466, -19018, -9138, -8530, -6402, -5946, -3970, -322, 2148, 8532, 9938, 10584, 14460, 21794, 25898, 26088, 28634.

 

m = 21;

smallest x:

23627359084542017700443765883390304876849497211964402702757103187986538276000593;

factors:

5903, 5923, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113;

offsets:

-55660, -53854, -50200, -47680, -44950, -38104, -28696, -25252, -18322, 2090, 5954, 15404, 16706, 30398, 34556, 34598, 37118, 40226, 44006, 46526, 55136.

 

m = 23;

smallest x:

60774447238043444828454541444960992901486086159678662537487565930308633350089334028209762683273263623;

factors:

24007, 24019, 24023, 24029, 24043, 24049, 24061, 24071, 24077, 24083, 24091, 24097, 24103, 24107, 24109, 24113, 24121, 24133, 24137, 24151, 24169, 24179, 24181;

offsets:

-75156, -65220, -57814, -57492, -53260, -52662, -45900, -26442, -20830, -15862, -13930, -13884, 10910, 15096, 28620, 34278, 36854, 41178, 53828, 53874, 67260, 71906, 84648.

 

m = 25;

smallest x:

740838664190452796966562537730861078765413428514892618665714733707102580576434966569421425040700787269;

factors:

11783, 11789, 11801, 11807, 11813, 11821, 11827, 11831, 11833, 11839, 11863, 11867, 11887, 11897, 11903, 11909, 11923, 11927, 11933, 11939, 11941, 11953, 11959, 11969, 11971;

offsets:

-62444, -60194, -56594, -53694, -49994, -42944, -29444, -21944, -15344, -8544, -4494, -494, 2506, 11256, 13656, 15756, 16806, 22956, 29806, 36756, 43806, 48156, 52056, 54556, 58056.

 

m = 27;

smallest x:

12780274072995637705351351322593908728841611166146035767498453457932330552191436057459407845862111339342001301009074914
115233;

factors:

39359, 39367, 39371, 39373, 39383, 39397, 39409, 39419, 39439, 39443, 39451, 39461, 39499, 39503, 39509, 39511, 39521, 39541, 39551, 39563, 39569, 39581, 39607, 39619, 39623, 39631, 39659;

offsets:

-93380, -74804, -74426, -72536, -67784, -56012, -55040, -50774, -45806, -39920, -12434, -10166, -9140, -5144, 4468, 13486, 30550, 30874, 37138, 54958, 55660, 56308, 57766, 58198, 67648, 95944, 104368.

 

m = 29;

smallest x:

268559041254834043074363841518581793103468552557923391422671177770175728209253368942523605789802892354099485010025
54270
9791589;

factors:

21013, 21017, 21019, 21023, 21031, 21059, 21061, 21067, 21089, 21101, 21107, 21121, 21139, 21143, 21149, 21157, 21163, 21169, 21179, 21187, 21191, 21193, 21211, 21221, 21227, 21247, 21269, 21277, 21283;

offsets:

-116830, -116308, -94152, -77970, -69966, -61092, -53320, -41778, -40212, -21072, -12430, -12198, -11908, 852, 16338, 17150, 24980, 25328, 28808, 38784, 49572, 53748, 57228, 58040, 63318, 65232, 70104, 77702, 82052.

***