A372891 Anti-elite primes (A128852) that are not prime factors of Fermat primes (A023394).

2, 13, 97, 193, 241, 673, 769, 2689, 5953, 8929, 12289, 40961, 49921, 61681, 101377, 286721, 414721, 417793, 550801, 786433, 1130641, 1376257, 1489153, 1810433, 3602561, 6942721, 7340033, 11304961, 12380161, 15790321, 17047297, 22253377, 39714817, 67411969, 89210881, 93585409, 113246209, 119782433, 152371201, 171048961, 185602561, 377487361, 394783681

Offset: 1

Author

Jianing Song, May 15 2024

Keywords

• nonn

Comments

Union of {2} and odd anti-elite primes p such that the multiplicative order of 2 modulo p is not a power of 2.

A128852 is the union of this sequence and prime factors of Fermat numbers >= 17.

Conjecture: All terms >= 97 are congruent to 1 modulo 8. (Note that every factor of Fermat numbers >= 17 is congruent to 1 modulo 8.)

Examples

			For n >= 2, we have 2^2^n + 1 == 4 (mod 13) for even n and 2^2^n + 1 == 10 (mod 13) for odd n. As 4 and 10 are both squares modulo 13, and 13 is not a factor of Fermat numbers (the multiplicative order of 2 modulo 13 is 12), 13 is a term.
For n >= 4, we have 2^2^n + 1 == 62 (mod 97) for even n and 2^2^n + 1 == 36 (mod 97) for odd n. As 36 and 62 are both squares modulo 97, and 97 is not a factor of Fermat numbers (the multiplicative order of 2 modulo 97 is 48), 97 is a term.
		

Crossrefs

Cf. A023394, A128852.

Programs

PARI

isA372891(n) = if(isprime(n) && n > 2, my(d = znorder(Mod(2, n))); if(isprimepower(2*d), return(0)); my(StartPoint = valuation(d, 2), LengthTest = znorder(Mod(2, d >> StartPoint))); for(i = StartPoint, StartPoint + LengthTest - 1, if(!issquare(Mod(2, n)^2^i + 1), return(0))); 1, n == 2)