Problems & Puzzles: Puzzles

Puzzle 1025. 352757

G. L. Honaker, Jr., made me notice this nice curio, by Metin Sariyar:

 The prime number 352757 divides the sum of reversals of the first 757253 primes [Sariyar]

Q. Is there another example like this?


Contributions during the week 25-31 Dec 2020, came from Emmanuel Vantieghem, Giorgos Kalogeropoulos, Oscar Volpatti, Jean-Marc REBERT, Paul Cleary, Simon Cavegn and Metin Sariyar.

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Emmanuel wrote:

The number of terms  n  in the sum is
    757253  when it's prime reverse divides the sum  8363564040709
    176282107  "    "     "             "        "          "     "  714971717854594723
    198616024   "    "     "            "        "          "     "  837804490805209436

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Giorgos wrote:

I found 2 more primes. So, here is the sequence: 
352757, 420616891, 701282671 

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Oscar wrote:

I understood puzzle 1025 as follows:
let a(n) be the sum of the reversals of the first n primes;
let q(n) be the reversal of n itself;
we want q(n) to be a prime factor of a(n).

 
I computed p(n), q(n), a(n) up to index n = 10^11:
p(10^11) = 2760727302517;
q(10^11) = 1;
a(10^11) = 361779481107064765431769.

 
I found 3 solutions with n not divisible by 10, so that n is the reversal of q(n) too.
I sorted them by ascending q(n):

 
a(757253) = 8363564040709 = 352757*23709137 (known);
a(198616024) = 837804490805209436 = 420616891*1991846996;
a(176282107) = 714971717854594723 = 701282671*1019520013 (emirp).

 
I also found 28 solutions with n divisible by 10, so that q(n) is much smaller than n itself.
I sorted them by ascending q(n), then by ascending n:

 
a(20) = 792 = 2*396;
a(200) = 233402 = 2*116701;
a(20000) = 6193952122 = 2*3096976061;
a(200000) = 706732091892 = 2*353366045946;
a(2000000) = 76716398744400 = 2*38358199372200;
a(200000000) = 845415962033088170 = 2*422707981016544085;
a(30) = 3456 = 3*1152
a(3000) = 103312083 = 3*34437361;
a(300000000) = 1395413008984700829 = 3*465137669661566943;
a(50) = 14900 = 5*2980;
a(5000) = 213520275 = 5*42704055;
a(500000) = 2356762606620 = 5*471352521324;
a(7000) = 323785280 = 7*46255040;
a(70000000000) = 196779500760183270057688 = 7*28111357251454752865384;
a(11000000000) = 3988789066613573674397 = 11*362617187873961243127;
a(7100000000) = 1843783459040396255306 = 17*108457850531788015018;
a(3500) = 130824988 = 53*2468396;
a(39100) = 16687699645 = 193*86464765;
a(9830) = 597427756 = 389*1535804;
a(90400) = 103883601215 = 409*253994135;
a(12400000) = 3934771203635270 = 421*9346249889870;
a(7880) = 372574593 = 887*420039;
a(30110) = 11753716905 = 1103*10656135;
a(760610) = 8546765263636 = 16067*531945308;
a(3996200) = 186496771039587 = 26993*6909079059;
a(3218200000) = 154221089516782761609 = 28123*5483806475723883;
a(725341000) = 17114129957494760866 = 143527*119239794306958;
a(79367230000) = 248299296385599276192989 = 3276397*75784252148197937.

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Jean-Marc wrorte:

Other exemples < 10^9

 
 The prime number 176 282 107 divides the sum of reversals of the first 701 282 671 primes.  
 The prime number 420 616 891 divides the sum of reversals of the first  198 616 024 primes.  

 
If leading zeros are permitted, I have found other exemples < 10^9, the reversals of :
20
30
50
200
3000
3500
5000
7000
7880
9830
20000
30110
39100
90400
200000
500000
757253
760610
2000000
3996200
12400000
176282107
198616024
200000000
300000000
725341000

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Paul wrote:

The Prime 701282671 divides the sum of the reversal of the first 176282107 primes.

 

714971717854594723/701282671 = 1019520013.

 

The Prime 420616891 divides the sum of the reversal of the first 198616024 primes.

 

837804490805209436/420616891 = 1991846996.

 

I didnít find anything else up to the sum of the first 3000000000 primes.

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Simon wrote:

The prime number 352757 divides the sum of reversals of the first 757253 primes.
The prime number 701282671 divides the sum of reversals of the first 176282107 primes.
The prime number 420616891 divides the sum of reversals of the first 198616024 primes.
Searched up to the first 308423464285 primes.

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Metin wrote:

New ones  I found are 701282671, 420616891.

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