OFFSET
1,1
COMMENTS
If p == 1 (mod 3) and p divides x^2 + x + 1, then p^2 divides (x+1)^p - x^p - 1; see A068209 for a proof.
Primes p == 1 (mod 3) such that A320535(primepi(p)) > 2.
Conjecture: this density of this sequence among the primes congruent to 1 modulo 3 is the same as that of A068209 among the primes congruent to 2 modulo 3. - Jianing Song, Nov 08 2022
LINKS
Jianing Song, Table of n, a(n) for n = 1..1312 (all terms up to 2*10^5)
EXAMPLE
For p = 79, the nontrivial solutions to (x+1)^p - x^p == 1 (mod p^2) are x == 11, 23, 32, 36, 42, 46, 55, 67 (mod 79). The equivalent classes x == 11, 32, 36, 42, 46, 67 (mod 79) satisfy x^2 + x + 1 != 0 (mod 79), so 79 is a term.
PROG
(PARI) isA358315(n) = if(isprime(n) && n%3==1, for(a=1, n-2, if(Mod(a+1, n^2)^n - Mod(a, n^2)^n==1 && znorder(Mod(a, n))!=3, return(1)))); return(0)
CROSSREFS
KEYWORD
nonn
AUTHOR
Jianing Song, Nov 08 2022
STATUS
approved