A060444 -id:A060444 - cp's OEIS frontend

A046799 Number of distinct prime factors of 2^n+1.

1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 3, 3, 1, 2, 4, 2, 2, 3, 3, 2, 3, 4, 4, 3, 2, 3, 5, 2, 2, 4, 4, 5, 4, 3, 4, 3, 2, 3, 6, 2, 3, 5, 5, 3, 3, 3, 5, 5, 3, 3, 6, 5, 3, 4, 3, 4, 4, 2, 5, 5, 2, 6, 6, 3, 3, 4, 6, 3, 5, 3, 5, 6, 4, 6, 9, 2, 3, 6, 5, 6, 5, 4, 5, 4, 4, 4, 10, 6, 2, 4, 4, 5, 3, 5, 6, 7, 6, 2, 9, 3, 2

Offset: 0

Labos Elemer

nonn

The length of row n in A060444.

			For n=7, 129 = 3.43 has 2 prime factors, so a(7) = 2.

Max Alekseyev, Table of n, a(n) for n = 0..1128 (terms 0..500 from T. D. Noe; terms 501..1062 from Amiram Eldar)

Cf. A000051, A001221, A060444, A086257 (number of primitive prime factors).

PrimeNu[1 + 2^#] & /@ Range[0, 104] (* Jayanta Basu, Jun 29 2013 *)

a(n) = omega(2^n+1); \\ Michel Marcus, Mar 25 2017

a(n) = A001221(A000051(n)). - Amiram Eldar, Oct 04 2019

A001269 Table T(n,k) in which n-th row lists prime factors of 2^n + 1 (n >= 0), with repetition.

Original entry on oeis.org

2, 3, 5, 3, 3, 17, 3, 11, 5, 13, 3, 43, 257, 3, 3, 3, 19, 5, 5, 41, 3, 683, 17, 241, 3, 2731, 5, 29, 113, 3, 3, 11, 331, 65537, 3, 43691, 5, 13, 37, 109, 3, 174763, 17, 61681, 3, 3, 43, 5419, 5, 397, 2113, 3, 2796203, 97, 257, 673, 3, 11, 251, 4051

Offset: 0

N. J. A. Sloane

Rows have irregular lengths.

The length of row n is A054992(n).

			Triangle begins:
  2;
  3;
  5;
  3,3,17;
  3,11;
  5,13;
  3,43;
  257;
  ...

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.

Max Alekseyev, Rows n = 0..1122, flattened (rows 0..500 from T. D. Noe)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
S. S. Wagstaff, Jr., The Cunningham Project
Chai Wah Wu, Tables from the Cunningham Project in machine-readable JSON format.

Cf. A060444 (factors w/o repetition), A054992 (row lengths).

repeat[{p_, e_}] := Table[p, {e}]; row[n_] := repeat /@ FactorInteger[2^n + 1] // Flatten; Table[row[n], {n, 0, 25}] // Flatten (* Jean-François Alcover, Jul 13 2012 *)

apply( A001269_row(n)=concat(apply(f->vector(f[2],i,f[1]), Col(factor(2^n+1))~)), [0..19]) \\ M. F. Hasler, Nov 19 2018

A069226 a(n) = gcd(n, 2^n + 1).

Original entry on oeis.org

2, 1, 1, 3, 1, 1, 1, 1, 1, 9, 5, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 27, 1, 1, 5, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 25, 3, 1, 1, 1, 11, 1, 3, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 17, 3, 5, 1, 1, 1, 1, 3, 1, 1, 13, 1, 1, 81, 1, 1, 1, 1, 1, 3, 1, 1, 5, 1, 1, 3, 1, 1, 1, 1, 1, 9, 1

Offset: 0

Vladeta Jovovic, Apr 12 2002

nonn

First occurrence of n: a(1)=1, a(3)=3, a(10)=5, a(9)=9, a(55)=11, a(78)=13, a(68)=17, a(50)=25, a(27)=27, a(406)=29, a(165)=33, a(666)=37, a(301)=43, a(1378)=53, a(1711)=59, a(390)=65, a(81)=81, a(3403)=83, a(2328)=97, a(495)=99, ... - R. J. Mathar, Dec 14 2016

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A006521 (fixed points), A014491, A049095, A049096, A060444.

Table[GCD[n,2^n+1],{n,100}] (* Harvey P. Dale, Dec 12 2012 *)

A069226(n) = gcd(n, 1+(1<Antti Karttunen, Jan 15 2025

Term a(0) = 2 prepended by Antti Karttunen, Jan 15 2025

cp's OEIS Frontend

A046799 Number of distinct prime factors of 2^n+1.

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A001269 Table T(n,k) in which n-th row lists prime factors of 2^n + 1 (n >= 0), with repetition.

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A069226 a(n) = gcd(n, 2^n + 1).

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