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.
***
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