Problems & Puzzles: Puzzles

Puzzle 31.- The Average Prime number, APN(k) = S(Pk)/k

Let’s define the "average prime number" APN(k) as S(Pk)/k, where S(Pk) is the sum of the first k prime numbers.

It happens that this number very few times is an integer. I have calculated the first five times it does:

 k P(k) S(pk) S(pk)/k 1 2 2 2 23 83 874 38 53 241 5830 110 853 6599 2615298 3066 11869 126551 712377380 60020

Can you get the following 3 cases ?

(see Puzzle 18 for similar questions about sum of consecutive primes)

Jo Yeong Uk (1/12/98) has found the following first three solutions to APN(k). Jack Brennen (21/05/99) found the fourth solution.

 k Pk S(Pk) S(Pk)/k Author(s) 117267 154479 86810649294 740282 Jo Yeong Uk 339615 4864121 794712005370 2340038 Jo Yeong Uk 3600489 60686737 105784534314378 29380602 Jo Yeong Uk 96643287 1966194317 92542301212047102 957565746 Jack Brennen

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Giovanni Resta wrote (Nov. 2004):

I found the following 2 new solutions.

i, pi, Si
2664167025, 63481708607, 82704567079549985700
43435512311, 1161468891953, 24733255676526572596026

No other solutions for pi < 4,011,201,392,413.

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On July 5, 2023, Paul W. Dyson wrote:

Last year I ran a program for about a month and a half on a RTX3060Ti GPU to find the next solution for S(pk) % pk = 0. This is Puzzle 18 question 2 on your Prime Puzzles website. The next value is pk = 55,691,042,365,834,801. This sequence is OEIS A007506.

The code was also searching for the next solution of S(pk) % k = 0 (i.e. % k, rather than % pk). So I kept it running until I found a solution to that one too. After a total of about 5 and a half months it found k = 6,361,476,515,268,337. This sequence is OEIS A045345.

I kept looking for S(pk) % pk = 0 for the whole time, and found that the next solution must be greater than pk = 253,814,097,223,614,463.

...

I've just seen that S(pk) % k = 0 is Puzzle 31. So I have a solution to that puzzle as well.

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