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A372891
Anti-elite primes (A128852) that are not prime factors of Fermat primes (A023394).
0
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
1,1
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.)
EXAMPLE
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.
PROG
(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)
CROSSREFS
Sequence in context: A104255 A369202 A118352 * A320360 A074614 A184019
KEYWORD
nonn
AUTHOR
Jianing Song, May 15 2024
STATUS
approved