Problems & Puzzles: Puzzles

Puzzle 1099 A Follow-up to puzzles 782 and 1097 Still inside of the field of prime producing functions, we present a new one sent by Simon Cavegn. Using integer division (rounded down), f(n) is prime for n=0..102 f(n)=41619623+n*103123020+n/13*(-1237476240)+n/26*(-65763840)+n/39*(-37359180)+n/65*2663271128 This interesting function produces 103 primes, in the range n=0 to 102. These are 4 primes unique, 15 primes that appears twice, and 23 primes that appears three times (4+15*2+23*3 = 4+30+69 = 103). Q. Can you find a better function of this type?

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During the week 12-18 August, 2002, contributions came from Emmanuel Vantieghem

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Emmanuel wrote:

This is a (longer but similar) formula that produces 120 distinct primes when  n = 0, 1, 2, ..., 119 :
 

  H[n] = 35969 + 9699690*n - Floor[n/10]*96801340 - Floor[n/20]*9796 - Floor[n/30]*22722

  + Floor[n/40]*1492548 + Floor[n/50]*148878 + Floor[n/60]*2555030 + Floor[n/70]*584054

  - Floor[n/80]*1540270 - Floor[n/90]*141506 + Floor[n/100]*2077168 + Floor[n/110*259696.

 

The produced primes are :
35969,9735659,19435349,29135039,38834729,48534419,58234109,67933799,77633489,87333179,

231529,9931219,19630909,29330599,39030289,48729979,58429669,68129359,77829049,875287390,

417293,10116983,19816673,29516363,39216053,48915743,58615433,68315123,78014813,87714503,

590131,10289821,19989511,29689201,39388891,49088581,58788271,68487961,78187651,87887341,

2268443,11968133,21667823,31367513,41067203,50766893,60466583,70166273,79865963,89565653,

2612881,12312571,22012261,31711951,41411641,51111331,60811021,70510711,80210401,89910091,

5330953,15030643,24730333,34430023,44129713,53829403,63529093,73228783,82928473,92628163,

6110567,15810257,25509947,35209637,44909327,54609017,64308707,74008397,83708087,93407777,

6248609,15948299,25647989,35347679,45047369,54747059,64446749,74146439,83846129,93545819,

6279941,15979631,25679321,35379011,45078701,54778391,64478081,74177771,83877461,93577151,

8691751,18391441,28091131,37790821,47490511,57190201,66889891,76589581,86289271,95988961,

9147007,18846697,28546387,38246077,47945767,57645457,67345147,77044837,86744527,96444217.

It may look impressive, but the "coefficients" are obtained by using the primes produced by this formula ...
 

As you can see, the primes are subdivided in twelve arithmetic progressions of length10, with difference  9699690.

I searched for them first, which took about 8 seconds.  If we take  n  from 0 to 9, all the terms containing "Floor" are zero.

If we take  n  from  10 to 19, only the first three terms play a role and form a linear function.

If we take n from 20 to 29, only the first four terms play a role and form a linear function, 

And so on ... 
 

In my opinion this is not a 'serious' prime function producer ...

 

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