cp's OEIS Frontend

A204620 Numbers k such that 3*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.

%I A204620 #64 Feb 16 2025 08:33:16
%S A204620 41,209,157169,213321,303093,382449,2145353,2478785
%N A204620 Numbers k such that 3*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.
%C A204620 Terms are odd: by Morehead's theorem, 3*2^(2*n) + 1 can never divide a Fermat number.
%C A204620 No other terms below 7516000.
%C A204620 Is this sequence the same as "Numbers k such that 3*2^k + 1 is a factor of a Fermat number 2^(2^m) + 1 for some m"? - _Arkadiusz Wesolowski_, Nov 13 2018
%C A204620 The last sentence of Morehead's paper is: "It is easy to show that _composite_ numbers of the forms 2^kappa * 3 + 1, 2^kappa * 5 + 1 can not be factors of Fermat's numbers." [a proof is needed]. - _Jeppe Stig Nielsen_, Jul 23 2019
%C A204620 Any factor of a Fermat number 2^(2^m) + 1 of the form 3*2^k + 1 is prime if k < 2*m + 6. - _Arkadiusz Wesolowski_, Jun 12 2021
%C A204620 If, for any m >= 0, F(m) = 2^(2^m) + 1 has a prime factor p of the form 3*2^k + 1, then F(m)/p is congruent to 11 mod 30. - _Arkadiusz Wesolowski_, Jun 13 2021
%C A204620 A number k belongs to this sequence if and only if the order of 2 modulo p is not divisible by 3, where p is a prime of the form 3*2^k + 1 (see Golomb paper). - _Arkadiusz Wesolowski_, Jun 14 2021
%H A204620 Solomon W. Golomb, <a href="https://doi.org/10.1090/S0025-5718-1976-0404129-8">Properties of the sequence 3.2^n+1</a>, Math. Comp., 30 (1976), 657-663.
%H A204620 Wilfrid Keller, <a href="http://www.prothsearch.com/fermat.html">Fermat factoring status</a>
%H A204620 J. C. Morehead, <a href="https://doi.org/10.1090/S0002-9904-1906-01371-4">Note on the factors of Fermat's numbers</a>, Bull. Amer. Math. Soc., Volume 12, Number 9 (1906), pp. 449-451.
%H A204620 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FermatNumber.html">Fermat Number</a>
%t A204620 lst = {}; Do[p = 3*2^n + 1; If[PrimeQ[p] && IntegerQ@Log[2, MultiplicativeOrder[2, p]], AppendTo[lst, n]], {n, 7, 209, 2}]; lst
%o A204620 (PARI) isok(n) = my(p = 3*2^n + 1, z = znorder(Mod(2, p))); isprime(p) && ((z >> valuation(z, 2)) == 1); \\ _Michel Marcus_, Nov 10 2018
%Y A204620 Subsequence of A002253.
%Y A204620 Cf. A000215, A039687, A057775, A057778, A201364, A226366.
%K A204620 nonn,hard,more
%O A204620 1,1
%A A204620 _Arkadiusz Wesolowski_, Jan 17 2012