A204620 -id:A204620 - cp's OEIS frontend

A226366 Numbers k such that 5*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.

7, 25, 39, 75, 127, 1947, 3313, 23473, 125413

Offset: 1

Arkadiusz Wesolowski, Jun 05 2013

No other terms below 5330000.

The reason all terms are odd is that if k is even, then 5*2^k + 1 == (-1)*(-1)^k + 1 = (-1)*1 + 1 = 0 (mod 3). So if k is even, then 3 divides 5*2^k + 1, and since 3 divides no other Fermat number than F_0=3 itself, we do not have a Fermat factor. - Jeppe Stig Nielsen, Jul 21 2019

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 A002254.

Cf. A000215, A050526, A057775, A057778, A201364, A204620.

lst = {}; Do[p = 5*2^n + 1; If[PrimeQ[p] && IntegerQ@Log[2, MultiplicativeOrder[2, p]], AppendTo[lst, n]], {n, 7, 3313, 2}]; lst

isok(n) = my(p = 5*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.

A229852 3*h^2, where h is an odd integer not divisible by 3.

Original entry on oeis.org

3, 75, 147, 363, 507, 867, 1083, 1587, 1875, 2523, 2883, 3675, 4107, 5043, 5547, 6627, 7203, 8427, 9075, 10443, 11163, 12675, 13467, 15123, 15987, 17787, 18723, 20667, 21675, 23763, 24843, 27075, 28227, 30603, 31827, 34347, 35643, 38307, 39675, 42483, 43923

Offset: 1

Arkadiusz Wesolowski, Oct 01 2013

If p = a(n)*2^k + 1 divides a composite Fermat number 2^(2^m) + 1 and p is a prime, then k is odd.

More precisely, k == 1 (mod 4) if h == +/- 1 (mod 5) and k == 3 (mod 4) if h == +/- 2 (mod 5) (Krizek, Luca and Somer).

M. Krizek, F. Luca, L. Somer, 17 Lectures on Fermat Numbers: From Number Theory to Geometry, CMS Books in Mathematics, vol. 9, Springer-Verlag, New York, 2001, pp. 63-65.

Colin Barker, Table of n, a(n) for n = 1..1000
Wilfrid Keller, Fermat factoring status.
Eric Weisstein's World of Mathematics, Fermat Number.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000215, A204620, A291050.

[3*h^2 : h in [1..121 by 2] | not IsZero(h mod 3)];

3*Select[Range[1, 121, 2], Mod[#, 3] > 0 &]^2 (* Amiram Eldar, Jan 02 2021 *)

forstep(h=1, 121, 2, if(!(h%3==0), print1(3*h^2, ", ")));

Vec(3*x*(1+24*x+22*x^2+24*x^3+x^4) / ((1-x)^3*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 26 2016

G.f.: 3*x*(1+24*x+22*x^2+24*x^3+x^4) / ((1-x)^3*(1+x)^2).
a(n) = 3*A104777(n).
From Colin Barker, Jan 26 2016: (Start)
a(n) = 3*(18*n^2+6*(-1)^n*n-18*n-3*(-1)^n+5)/2.
a(n) = 27*n^2-18*n+3 for n even.
a(n) = 27*n^2-36*n+12 for n odd.
(End)
Sum_{n>=1} 1/a(n) = Pi^2/27 (A291050). - Amiram Eldar, Jan 02 2021

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A226366 Numbers k such that 5*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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A229852 3*h^2, where h is an odd integer not divisible by 3.

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