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Anti-elite primes (A128852) that are not prime factors of Fermat primes (A023394).
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%I #7 May 19 2024 14:35:35

%S 2,13,97,193,241,673,769,2689,5953,8929,12289,40961,49921,61681,

%T 101377,286721,414721,417793,550801,786433,1130641,1376257,1489153,

%U 1810433,3602561,6942721,7340033,11304961,12380161,15790321,17047297,22253377,39714817,67411969,89210881,93585409,113246209,119782433,152371201,171048961,185602561,377487361,394783681

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

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

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

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

%e 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.

%e 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.

%o (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)

%Y Cf. A023394, A128852.

%K nonn

%O 1,1

%A _Jianing Song_, May 15 2024