A226366 -id:A226366 - 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.

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

Offset: 1

Arkadiusz Wesolowski, Jan 17 2012

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

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

Subsequence of A002253.

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

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

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

A268657 Numbers k such that 3*2^k + 1 is a prime factor of a generalized Fermat number 3^(2^m) + 1 for some m.

Original entry on oeis.org

6, 12, 18, 30, 36, 41, 66, 189, 201, 209, 276, 408, 438, 534, 2208, 3168, 3189, 3912, 34350, 42294, 44685, 48150, 54792, 55182, 59973, 80190, 157169, 213321, 303093, 382449, 709968, 801978, 916773, 1832496, 2145353, 2291610, 2478785, 5082306, 7033641, 10829346

Offset: 1

Arkadiusz Wesolowski, Feb 10 2016

nonn

Wilfrid Keller, private communication, 2008.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..41
Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
C. K. Caldwell, Top Twenty page, Generalized Fermat Divisors (base=3)
OEIS Wiki, Generalized Fermat numbers

Cf. A059919, A268658, A204620, A268659, A268660, A268661, A268662, A268663, A226366, A268664. Subsequence of A002253.

for(k=1,+oo,p=3*2^k+1;if(ispseudoprime(p),t=znorder(Mod(3,p));bitand(t,t-1)==0&&print1(k,", "))) \\ Jeppe Stig Nielsen, Oct 30 2020

A268658 Numbers k such that 3*2^k + 1 is a prime factor of a generalized Fermat number 5^(2^m) + 1 for some m.

Original entry on oeis.org

2, 8, 18, 66, 189, 209, 408, 2208, 2816, 3168, 3912, 20909, 54792, 59973, 157169, 303093, 709968, 801978, 1832496, 2145353, 2291610, 5082306, 10829346, 16408818

Offset: 1

Arkadiusz Wesolowski, Feb 10 2016

Wilfrid Keller, private communication, 2008.

Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
C. K. Caldwell, Top Twenty page, Generalized Fermat Divisors (base=5)
OEIS Wiki, Generalized Fermat numbers

Cf. A199591, A268657, A204620, A268659, A268660, A268661, A268662, A268663, A226366, A268664. Subsequence of A002253.

for(k=1,+oo,p=3*2^k+1;if(ispseudoprime(p),t=znorder(Mod(5,p));bitand(t,t-1)==0&&print1(k,", "))) \\ Jeppe Stig Nielsen, Oct 30 2020

a(24) from Jeppe Stig Nielsen, Oct 30 2020

A268660 Numbers n such that 3*2^n + 1 is a prime factor of a generalized Fermat number 12^(2^m) + 1 for some m.

Original entry on oeis.org

2, 5, 8, 41, 209, 353, 2816, 20909, 42665, 157169, 213321, 303093, 362765, 382449, 2145353, 2478785

Offset: 1

Arkadiusz Wesolowski, Feb 10 2016

Wilfrid Keller, private communication, 2008.

Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
C. K. Caldwell, Top Twenty page, Generalized Fermat Divisors (base=12)
OEIS Wiki, Generalized Fermat numbers

Cf. A152585, A268657, A268658, A204620, A268659, A268661, A268662, A268663, A226366, A268664. Subsequence of A002253.

A268661 Numbers n such that 5*2^n + 1 is a prime factor of a generalized Fermat number 3^(2^m) + 1 for some m.

Original entry on oeis.org

3, 55, 127, 13165, 240937, 819739, 1282755

Offset: 1

Arkadiusz Wesolowski, Feb 10 2016

Wilfrid Keller, private communication, 2008.

Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
C. K. Caldwell, Top Twenty page, Generalized Fermat Divisors (base=3)
OEIS Wiki, Generalized Fermat numbers

Cf. A059919, A268662, A268663, A226366, A268664, A268657, A268658, A204620, A268659, A268660. Subsequence of A002254.

A268662 Numbers n such that 5*2^n + 1 is a prime factor of a generalized Fermat number 5^(2^m) + 1 for some m.

Original entry on oeis.org

7, 15, 25, 39, 55, 75, 85, 127, 1947, 3313, 13165, 23473, 125413, 1282755, 1777515

Offset: 1

Arkadiusz Wesolowski, Feb 10 2016

Wilfrid Keller, private communication, 2008.

Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
C. K. Caldwell, Top Twenty page, Generalized Fermat Divisors (base=5)
OEIS Wiki, Generalized Fermat numbers

Cf. A199591, A268661, A268663, A226366, A268664, A268657, A268658, A204620, A268659, A268660. Subsequence of A002254.

A268663 Numbers n such that 5*2^n + 1 is a prime factor of a generalized Fermat number 6^(2^m) + 1 for some m.

Original entry on oeis.org

127, 4687, 1777515

Offset: 1

Arkadiusz Wesolowski, Feb 10 2016

Wilfrid Keller, private communication, 2008.

Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
C. K. Caldwell, Top Twenty page, Generalized Fermat Divisors (base=6)
OEIS Wiki, Generalized Fermat numbers

Cf. A078303, A268661, A268662, A226366, A268664, A268657, A268658, A204620, A268659, A268660. Subsequence of A002254.

A268664 Numbers n such that 5*2^n + 1 is a prime factor of a generalized Fermat number 12^(2^m) + 1 for some m.

Original entry on oeis.org

13, 15, 127, 5947, 26607, 1320487

Offset: 1

Arkadiusz Wesolowski, Feb 10 2016

Wilfrid Keller, private communication, 2008.

Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
C. K. Caldwell, Top Twenty page, Generalized Fermat Divisors (base=12)
OEIS Wiki, Generalized Fermat numbers

Cf. A152585, A268661, A268662, A268663, A226366, A268657, A268658, A204620, A268659, A268660. Subsequence of A002254.

A268659 Numbers n such that 3*2^n + 1 is a prime factor of a generalized Fermat number 10^(2^m) + 1 for some m.

Original entry on oeis.org

209, 44685, 157169, 303093, 362765, 916773, 2145353

Offset: 1

Arkadiusz Wesolowski, Feb 10 2016

Wilfrid Keller, private communication, 2008.

Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
C. K. Caldwell, Top Twenty page, Generalized Fermat Divisors (base=10)
OEIS Wiki, Generalized Fermat numbers

Cf. A080176, A268657, A268658, A204620, A268660, A268661, A268662, A268663, A226366, A268664. Subsequence of A002253.

A280003 Numbers k such that 7*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.

Original entry on oeis.org

14, 120, 290, 320, 95330, 2167800

Offset: 1

Arkadiusz Wesolowski, Feb 21 2017

18233956 belongs to this sequence, but its position is currently unknown. - Jeppe Stig Nielsen, Oct 05 2020

Proth Search Page, Prime factors of Fermat numbers
Eric Weisstein's World of Mathematics, Fermat Number

Subsequence of A032353.

Cf. A000215, A050527, A057778, A201364, A204620, A226366, A280004.

IsInteger := func; [n: n in [1..320] | IsPrime(k) and IsInteger(Log(2, Modorder(2, k))) where k is 7*2^n+1];

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.

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A268657 Numbers k such that 3*2^k + 1 is a prime factor of a generalized Fermat number 3^(2^m) + 1 for some m.

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A268658 Numbers k such that 3*2^k + 1 is a prime factor of a generalized Fermat number 5^(2^m) + 1 for some m.

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A268660 Numbers n such that 3*2^n + 1 is a prime factor of a generalized Fermat number 12^(2^m) + 1 for some m.

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A268661 Numbers n such that 5*2^n + 1 is a prime factor of a generalized Fermat number 3^(2^m) + 1 for some m.

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A268662 Numbers n such that 5*2^n + 1 is a prime factor of a generalized Fermat number 5^(2^m) + 1 for some m.

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A268663 Numbers n such that 5*2^n + 1 is a prime factor of a generalized Fermat number 6^(2^m) + 1 for some m.

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A268664 Numbers n such that 5*2^n + 1 is a prime factor of a generalized Fermat number 12^(2^m) + 1 for some m.

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A268659 Numbers n such that 3*2^n + 1 is a prime factor of a generalized Fermat number 10^(2^m) + 1 for some m.

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A280003 Numbers k such that 7*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.

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