A372894 A positive integer 2^k*m, where m is odd and k >= 0, belongs to this sequence iff the Jacobi symbol (F_n/m) = -1 for only a finite number of Fermat numbers F_n = A000215(n).

1, 2, 4, 8, 9, 13, 15, 16, 17, 18, 21, 25, 26, 30, 32, 34, 35, 36, 42, 49, 50, 52, 60, 64, 68, 70, 72, 81, 84, 97, 98, 100, 104, 117, 120, 121, 123, 128, 135, 136, 140, 144, 153, 162, 168, 169, 189, 193, 194, 195, 196, 200, 205, 208, 221, 225, 234, 240, 241, 242, 246, 255, 256, 257, 270, 272, 273, 280, 287, 288, 289

Offset: 1

Author

Jianing Song, May 15 2024

Keywords

• nonn

Comments

Can be seen as the opposite of A129802.

Let m be an odd number and ord(2,m) = 2^r*d be the multiplicative order of 2 modulo m, where d is odd, then 2^2^n + 1 is congruent to one of 2^2^r + 1, 2^2^(r+1) + 1, ..., 2^2^(r+ord(2,d)-1) + 1 modulo m, so it suffices to check these ord(2,d) numbers.

Note that if m > 1, then m does not divide 2^2^n + 1 for n >= r, otherwise we would have 2^(2^n*d) = (2^ord(2,m))^2^(n-r) == 1 (mod m) and 2^(2^n*d) = (2^2^n)^d == (-1)^d == -1 (mod m). As a result, m is a term if and only if the Jacobi symbol ((2^2^n + 1)/m) is equal to 1 for m = r, r+1, ..., r+ord(2,d)-1.

By definition, a squarefree number that is a product of elite primes (A102742) or anti-elite primes (A128852) is a term if and only if its number of elite factors is even. But a squarefree term can have factors that are neither elite nor anti-elite, the smallest being 341 = 11*31.

Contains divisors of Fermat numbers >= 17 (A307843 \ {3,5}) since they are products of elite primes.

Examples

			For n >= 1, we have 2^2^n + 1 == 65, 5, 17, 257 (mod 341) respectively for n == 0, 1, 2, 3 (mod 4). As we have (65/341) = (5/341) = (17/341) = (257/341) = 1, 341 is a term.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000215, A102742, A129802, A307843.

Prime elements of this sequence are given by A128852.

Programs

PARI

isA372894(n) = n = (n >> valuation(n,2)); my(d = znorder(Mod(2, n)), StartPoint = valuation(d, 2), LengthTest = znorder(Mod(2, d >> StartPoint))); for(i = StartPoint, StartPoint + LengthTest - 1, if(kronecker(lift(Mod(2, n)^2^i + 1), n) == -1, return(0))); 1