A268658 -id:A268658 - cp's OEIS frontend

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.

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

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.

A273946 Odd prime factors of generalized Fermat numbers of the form 5^(2^m) + 1 with m >= 0.

Original entry on oeis.org

3, 13, 17, 257, 313, 641, 769, 2593, 11489, 19457, 65537, 163841, 786433, 1503233, 1655809, 7340033, 14155777, 18395137, 23606273, 29423041, 39714817, 75068993, 167772161, 2483027969, 4643094529, 6616514561, 47148957697, 241931001601, 2748779069441

Offset: 1

Arkadiusz Wesolowski, Jun 05 2016

nonn

Odd primes p such that the multiplicative order of 5 (mod p) is a power of 2.

Hans Riesel, Common prime factors of the numbers A_n=a^(2^n)+1, BIT 9 (1969), pp. 264-269.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..37
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)
Harvey Dubner and Wilfrid Keller, Factors of Generalized Fermat Numbers, Math. Comp. 64 (1995), no. 209, pp. 397-405.
OEIS Wiki, Generalized Fermat numbers

Cf. A023394, A072982, A199591, A268658, A268662, A273945 (base 3), A273947 (base 6), A273948 (base 7), A273949 (base 11), A273950 (base 12).

Select[Prime@Range[2, 10^5], IntegerQ@Log[2, MultiplicativeOrder[5, #]] &]

A282943 Numbers k such that 3*2^k + 1 is a prime factor of a generalized Fermat number 7^(2^m) + 1 for some m.

Original entry on oeis.org

8, 12, 36, 276, 408, 2208, 2816, 3168, 3912, 42665, 44685, 59973, 709968, 916773, 1832496

Offset: 1

Arkadiusz Wesolowski, Feb 25 2017

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.
OEIS Wiki, Generalized Fermat numbers

Cf. A078304, A204620, A268657, A268658, A268659, A282944, A268660.

Subsequence of A002253.

SetDefaultRealField(RealField(400)); IsInteger := func; [n: n in [2..408] | IsPrime(k) and IsInteger(Log(2, Modorder(7, k))) where k is 3*2^n+1];

lst = {}; Do[p = 3*2^n + 1; If[PrimeQ[p] && IntegerQ@Log[2, MultiplicativeOrder[7, p]], AppendTo[lst, n]], {n, 3912}]; lst

A282944 Numbers k such that 3*2^k + 1 is a prime factor of a generalized Fermat number 11^(2^m) + 1 for some m.

Original entry on oeis.org

6, 30, 36, 66, 276, 353, 2816, 3189, 34350, 48150, 80190, 1832496, 2291610, 5082306, 10829346

Offset: 1

Arkadiusz Wesolowski, Feb 25 2017

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.
OEIS Wiki, Generalized Fermat numbers

Cf. A199592, A204620, A268657, A268658, A282943, A268659, A268660.

Subsequence of A002253.

SetDefaultRealField(RealField(350)); IsInteger := func; [n: n in [2..353] | IsPrime(k) and IsInteger(Log(2, Modorder(11, k))) where k is 3*2^n+1];

lst = {}; Do[p = 3*2^n + 1; If[PrimeQ[p] && IntegerQ@Log[2, MultiplicativeOrder[11, p]], AppendTo[lst, n]], {n, 3189}]; lst

cp's OEIS Frontend

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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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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A273946 Odd prime factors of generalized Fermat numbers of the form 5^(2^m) + 1 with m >= 0.

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A282943 Numbers k such that 3*2^k + 1 is a prime factor of a generalized Fermat number 7^(2^m) + 1 for some m.

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A282944 Numbers k such that 3*2^k + 1 is a prime factor of a generalized Fermat number 11^(2^m) + 1 for some m.

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Magma

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