A274994 Primes p such that p^2 divides Sum_{k=1..(p-1)/2} (k^(p-2))*(k^(p-1)-1).

3, 1093, 3511, 9511, 13691

Offset

1,1

Comments

Any prime p divides Sum_{k=1..(p-1)/2} (k^(p-2))*(k^(p-1)-1). But a restricted list of primes p are such that p^2 divides Sum_{k=1..(p-1)/2}(k^(p-2))*(k^(p-1)-1).

Also primes p such that (2^(p-1)-1)/p == 0 (mod p) or 2*((p-1)!+1)/p +(2^(p-1)-1)/p == 0 (mod p), because it can be shown that Sum_{k=1..(p-1)/2} (k^(p-2))*(k^(p-1)-1) == p*((2^(p-1)-1)/p)*(2*((p-1)!+1)/p +(2^(p-1)-1)/p) (mod p^2).

The Wieferich primes (A001220) belong to the sequence.

No more terms up to 2000000, because A280300 has no more terms up to 2000000, and A001220 has no other terms below 4.97*10^17 (see the comments in these sequences). - René Gy, Jan 01 2017

Links

Table of n, a(n) for n=1..5.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.

Mathematica

p=3; While[p<20000, If[Mod[Sum[PowerMod[k, p-2, p^2]*(PowerMod[k, p-1, p^2]-1), {k, 1, (p-1)/2}], p^2] == 0, Print [p]]; p=NextPrime[p]]

Prog

(PARI) is(n)=if(!isprime(n), return(0)); my(m=n^2, e=n-2); sum(k=1, n\2, Mod(k, m)^e*(Mod(k, m)^(e+1)-1))==0 && n>2 \\ Charles R Greathouse IV, Nov 13 2016

Crossrefs

Equals the union of A001220 and A280300.

Sequence in context: A065604 A062612 A096082 * A178195 A199236 A171360

Adjacent sequences: A274991 A274992 A274993 * A274995 A274996 A274997

Keyword

nonn,more

Author

René Gy, Nov 11 2016

Status

approved