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

41, 209, 157169, 213321, 303093, 382449, 2145353, 2478785

Offset

1,1

Comments

Terms are odd: by Morehead's theorem, 3*2^(2*n) + 1 can never divide a Fermat number.

No other terms below 7516000.

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

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

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

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

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

Links

Table of n, a(n) for n=1..8.

Solomon W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663.

Wilfrid Keller, Fermat factoring status

J. C. Morehead, Note on the factors of Fermat's numbers, Bull. Amer. Math. Soc., Volume 12, Number 9 (1906), pp. 449-451.

Eric Weisstein's World of Mathematics, Fermat Number

Mathematica

lst = {}; Do[p = 3*2^n + 1; If[PrimeQ[p] && IntegerQ@Log[2, MultiplicativeOrder[2, p]], AppendTo[lst, n]], {n, 7, 209, 2}]; lst

Prog

(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

Crossrefs

Subsequence of A002253.

Cf. A000215, A039687, A057775, A057778, A201364, A226366.

Sequence in context: A142526 A088319 A297598 * A172085 A251094 A300464

Adjacent sequences: A204617 A204618 A204619 * A204621 A204622 A204623

Keyword

nonn,hard,more

Author

Arkadiusz Wesolowski, Jan 17 2012

Status

approved