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 
***
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.
***
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.
***
