A002586 -id:A002586 - cp's OEIS frontend

A054992 Number of prime factors of 2^n + 1 (counted with multiplicity).

1, 1, 2, 1, 2, 2, 2, 1, 4, 3, 2, 2, 2, 3, 4, 1, 2, 4, 2, 2, 4, 3, 2, 3, 4, 4, 6, 2, 3, 6, 2, 2, 5, 4, 5, 4, 3, 4, 4, 2, 3, 6, 2, 3, 7, 5, 3, 3, 3, 7, 6, 3, 3, 6, 6, 3, 5, 3, 4, 4, 2, 5, 7, 2, 6, 6, 3, 4, 5, 7, 3, 5, 3, 5, 7, 4, 6, 10, 2, 3, 10, 5, 6, 5, 4, 5, 5, 4, 4, 11, 6, 2, 5, 4, 5, 3, 5, 6, 9, 6, 2, 9, 3

Offset: 1

Arne Ring (arne.ring(AT)epost.de), May 30 2000

nonn

The length of row n in A001269.

			a(3) = 2 because 2^3 + 1 = 9 = 3*3.

Max Alekseyev, Table of n, a(n) for n = 1..1128
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n+1): A057934 (b=10), A057935 (b=9), A057936 (b=8), A057937 (b=7), A057938 (b=6), A057939 (b=5), A057940 (b=4), A057941 (b=3), this sequence (b=2).
Cf. A000051, A002586, A002587, A003260, A001222, A001269, A001348, A054988, A054989, A054990, A054991, A000978.
Cf. A046051 (number of prime factors of 2^n-1).
Cf. A086257 (number of primitive prime factors).

a[n_] := Module[{x=FactorInteger[2^n+1]}, Sum[x[[i]][[2]], {i, Length[x]}]]
A054992[n_Integer] := PrimeOmega[2^n + 1]; Table[A054992[n], {n,103}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)

a(n)=bigomega(2^n+1) \\ Charles R Greathouse IV, Apr 29 2015

a(n) = A046051(2n) - A046051(n). - T. D. Noe, Jun 18 2003

a(n) = A001222(A000051(n)). - Amiram Eldar, Oct 04 2019

Extended by Patrick De Geest, Oct 01 2000
Terms to a(500) in b-file from T. D. Noe, Nov 10 2007
Deleted duplicate (and broken) Wagstaff link. - N. J. A. Sloane, Jan 18 2019
a(500)-a(1062) in b-file from Amiram Eldar, Oct 04 2019
a(1063)-a(1128) in b-file from Max Alekseyev, Jul 15 2023, Mar 15 2025

A002587 Largest prime factor of 2^n + 1.

Original entry on oeis.org

2, 3, 5, 3, 17, 11, 13, 43, 257, 19, 41, 683, 241, 2731, 113, 331, 65537, 43691, 109, 174763, 61681, 5419, 2113, 2796203, 673, 4051, 1613, 87211, 15790321, 3033169, 1321, 715827883, 6700417, 20857, 26317, 86171, 38737, 25781083, 525313

Offset: 0

N. J. A. Sloane

nonn

a(n) != 1 (mod n) for n = 3, 51, 141, 309, 321, 348, ... - Giovanni Resta & Thomas Ordowski, Jan 05 2014

a(n) != 1 (mod n) iff a(m) = a(n) for some m < n. Then n = 3m for m = 1, 17, 47, 103, 107, 116, ... - Thomas Ordowski, Jan 08 2014

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 2, p. 85.
E. Lucas, Théorie des fonctions numériques simplement périodiques, Amer. J. Math., 1 (1878), 184-239, 289-321.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Max Alekseyev, Table of n, a(n) for n = 0..1128 (terms 1..500 from T. D. Noe, terms 1037..1062 from Amiram Eldar, term 1108 from Tyler Busby)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
D. X. Charles, The abc-conjecture and the largest prime factor of 2^n + 1
Edouard Lucas, Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240, 289-321. See pages 239 and 240.
Edouard Lucas, The Theory of Simply Periodic Numerical Functions, Fibonacci Association, 1969. English translation of article "Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240.
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A000051, A002586, A006530.

Cf. similar sequences listed in A274903.

[Maximum(PrimeDivisors(2^n+1)): n in [0..40]]; // Vincenzo Librandi, Jul 12 2016

Table[FactorInteger[2^n + 1][[-1, 1]], {n, 0, 30}] (* Vincenzo Librandi, Jul 12 2016 *)

a(n)=my(f=factor(2^n+1)[,1]); f[#f] \\ Charles R Greathouse IV, Jul 12 2016

Charles proves that a(n) >> n^(4/3) infinitely often under the abc conjecture, and reports that Andrew Granville has improved this to a(n) >> n^2. - Charles R Greathouse IV, Apr 29 2013

a(n) = A006530(A000051(n)). - Vincenzo Librandi, Jul 12 2016

More terms from James Sellers, Jul 06 2000

Offset 0, a(0) = 2 from Vincenzo Librandi, Jul 12 2016

A066135 a(n) = least number m > 1 such that sigma_n(m) = k*m for some k.

Original entry on oeis.org

6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 194, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 228, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 194, 6, 10, 6

Offset: 1

Labos Elemer, Dec 06 2001

nonn

a(n) <= 2p, where p = A002586(n) is the smallest prime factor of (1 + 2^n). (Proof. Since sigma_n(2p) = (1 + 2^n)(1 + p^n) and p is odd, 2p divides sigma_n(2p).) - Jonathan Sondow, Nov 23 2012

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000203, A001157, A001158, A001159, A002586, A007691, A046762-A046764, A055709-A055717.

Cf. A218860, A218861 (unique values and where they first occur).

Table[m = 2; While[Mod[DivisorSigma[n, m], m] > 0, m++]; m, {n, 100}] (* T. D. Noe, Nov 23 2012 *)

Sum{d^n} = ka(n), d runs over the divisors of a(n), where k is an integer and a(n) is the smallest suitable number.

Definition and formulas corrected by Jonathan Sondow, Nov 23 2012

A366719 Smallest prime dividing 12^n + 1.

Original entry on oeis.org

2, 13, 5, 7, 89, 13, 5, 13, 17, 7, 5, 13, 89, 13, 5, 7, 153953, 13, 5, 13, 41, 7, 5, 13, 17, 13, 5, 7, 89, 13, 5, 13, 769, 7, 5, 13, 89, 13, 5, 7, 17, 13, 5, 13, 89, 7, 5, 13, 7489, 13, 5, 7, 89, 13, 5, 13, 17, 7, 5, 13, 41, 13, 5, 7, 36097, 13, 5, 13, 89, 7

Offset: 0

Sean A. Irvine, Oct 17 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..511

Cf. A002586, A038371, A178248, A366712, A366713, A366714, A366715, A366716, A366717, A366720.

Table[FactorInteger[12^n + 1][[1,1]], {n, 0, 69}] (* Paul F. Marrero Romero, Oct 25 2023 *)

a(n) = A020639(A178248(n)). - Paul F. Marrero Romero, Oct 25 2023

A366609 Smallest prime dividing 4^n + 1.

Original entry on oeis.org

2, 5, 17, 5, 257, 5, 17, 5, 65537, 5, 17, 5, 97, 5, 17, 5, 641, 5, 17, 5, 257, 5, 17, 5, 193, 5, 17, 5, 257, 5, 17, 5, 274177, 5, 17, 5, 97, 5, 17, 5, 65537, 5, 17, 5, 257, 5, 17, 5, 641, 5, 17, 5, 257, 5, 17, 5, 449, 5, 17, 5, 97, 5, 17, 5, 59649589127497217

Offset: 0

Sean A. Irvine, Oct 14 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..1983

Bisection of A002586.

Cf. A002586, A038371, A052539, A074476, A274903, A366605, A366606, A366607, A366608, A366670, A366671, A366719.

A366670 Smallest prime dividing 6^n + 1.

Original entry on oeis.org

2, 7, 37, 7, 1297, 7, 13, 7, 17, 7, 37, 7, 1297, 7, 37, 7, 353, 7, 13, 7, 41, 7, 37, 7, 17, 7, 37, 7, 281, 7, 13, 7, 2753, 7, 37, 7, 577, 7, 37, 7, 17, 7, 13, 7, 89, 7, 37, 7, 193, 7, 37, 7, 1297, 7, 13, 7, 17, 7, 37, 7, 41, 7, 37, 7, 4926056449, 7, 13, 7, 137

Offset: 0

Sean A. Irvine, Oct 15 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..4095

Cf. A020639, A062394, A274904, A366630.

Cf. A002586, A366609, A366671, A038371, A366719.

Table[FactorInteger[6^n + 1][[1,1]], {n, 0, 68}] (* Paul F. Marrero Romero, Oct 17 2023 *)

a(n) = A020639(A062394(n)). - Paul F. Marrero Romero, Oct 17 2023

A066284 a(n) = A066135(4*n).

Original entry on oeis.org

34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 228, 34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 228, 34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 386, 34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 228, 34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 228, 34, 84, 34, 84, 34, 194

Offset: 1

Labos Elemer, Dec 11 2001

nonn

a(n) <= 2p, where p = A002586(4n) is the least prime factor of (1 + 16^n). (See the Comment in A066135.) - Jonathan Sondow, Nov 23 2012

			First 3 terms correspond to entries of other sequences as follows: a(1)=A046763(2), a(2)=A055712(2), a(3)=A055716(2).
From _Michael De Vlieger_, Jul 17 2017: (Start)
First position of values, with observations pertaining to values for 1 <= n <= 3000:
    Value   Position   Observations:
    --------------------------------
       34     1        All odd.
       84     2        In A047235.
      194     6        In A017593.
      228    12
      386    36
     1282    72
     1538   144
     3084   288
   147468   576
     1956   864
  1046532  1152
    24578  2304
     3252  2880
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..1000

Cf. A066135, A046761, A046762, A046763, A055709-A055717, A007691, A001159.

Table[m = 2; While[Mod[DivisorSigma[4 n, m], m] > 0, m++]; m, {n, 66}] (* Michael De Vlieger, Jul 17 2017 *)

a(n) = {n *= 4; my(m = 2); while (sigma(m, n) % m, m++); m;} \\ Michel Marcus, Oct 02 2016

a(n) = Min{x : sigma_4n(x) mod x = 0, x > 1}

A366671 Smallest prime dividing 8^n + 1.

Original entry on oeis.org

2, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 193, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 641, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 193, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 769, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5

Offset: 0

Sean A. Irvine, Oct 15 2023

nonn

a(n) = 3 if n is odd. a(n) = 5 if n == 2 (mod 4). - Robert Israel, Nov 20 2023

Max Alekseyev, Table of n, a(n) for n = 0..16383

Cf. A002586, A366609, A052536, A366655, A274905, A366670, A038371, A366719.

P1000:= mul(ithprime(i),i= 4..1000):
f:= proc(n) local t;
  if n::odd then return 3 elif n mod 4 = 2 then return 5 fi;
  t:= igcd(8^n+1,P1000);
  if t <> 1 then min(numtheory:-factorset(t)) else min(numtheory:-factorset(8^n+1)) fi
end proc:
map(f, [$0..100]); # Robert Israel, Nov 20 2023

Table[FactorInteger[8^n + 1][[1,1]], {n, 0, 78}] (* Paul F. Marrero Romero, Oct 20 2023 *)

from sympy import primefactors
def A366671(n): return min(primefactors((1<<3*n)+1)) # Chai Wah Wu, Oct 16 2023

a(n) = A020639(A062395(n)). - Paul F. Marrero Romero, Oct 20 2023

a(n) = A002586(3*n) for n >= 1. - Robert Israel, Nov 20 2023

A374237 Irregular triangle read by rows where row n lists, in increasing order, the divisors of 2^n + 1.

Original entry on oeis.org

1, 2, 1, 3, 1, 5, 1, 3, 9, 1, 17, 1, 3, 11, 33, 1, 5, 13, 65, 1, 3, 43, 129, 1, 257, 1, 3, 9, 19, 27, 57, 171, 513, 1, 5, 25, 41, 205, 1025, 1, 3, 683, 2049, 1, 17, 241, 4097, 1, 3, 2731, 8193, 1, 5, 29, 113, 145, 565, 3277, 16385, 1, 3, 9, 11, 33, 99, 331, 993, 2979, 3641, 10923, 32769

Offset: 0

Paolo Xausa, Jul 02 2024

			Triangle begins:
   [0]  1,   2;
   [1]  1,   3;
   [2]  1,   5;
   [3]  1,   3,  9;
   [4]  1,  17;
   [5]  1,   3, 11,  33;
   [6]  1,   5, 13,  65;
   [7]  1,   3, 43, 129;
   [8]  1, 257;
   [9]  1,   3,  9,  19,  27,   57, 171, 513;
  [10]  1,   5, 25,  41, 205, 1025;
  ...

Paolo Xausa, Table of n, a(n) for n = 0..11484 (rows 0..135 of the triangle, flattened).
Index entries for sequences related to divisors of numbers

Subsequence of A027750.
Cf. A000051, A002586 (2nd column), A046798 (row lengths), A069060 (row products), A069061 (row sums).
Cf. A361438 (analogous for 2^n - 1).

T:= n-> sort([numtheory[divisors](2^n+1)[]])[]:
seq(T(n), n=0..15);  # Alois P. Heinz, Oct 20 2024

Mathematica
```
Divisors[2^Range[0, 20] + 1]
```

A292834 Numbers m, not powers of 2, such that the least prime factor of 2^m + 1 is congruent to 1 (mod m).

Original entry on oeis.org

24, 48, 112, 160, 192, 272, 448, 496, 656, 688, 832, 896, 1088, 1152, 1168, 1328, 1360, 1408, 1472, 1520, 1664, 1744, 1920, 1984, 2176, 2304, 2432, 2560, 2688, 2752, 2816, 2944, 2960, 3056, 3072, 3200, 3328, 3520, 3664, 3712, 3776, 4672, 4864, 4928, 5120, 5376, 5552, 5888, 6144

Offset: 1

Thomas Ordowski, Sep 24 2017

Problem: are there infinitely many such numbers?
Theorem: there are no numbers m in the sequence such that, for each prime factor p of 2^m + 1, p == 1 (mod m).
Proof: if all prime factors p of 2^m + 1 are p == 1 (mod m), then 2^m + 1 == 1 (mod m), thus 2^m == 0 (mod m), so m = 2^k.
From Theorem in A002586, all terms are == 0 (mod 8). - Robert G. Wilson v, Jan 02 2018

Robert G. Wilson v, Table of n, a(n) for n = 1..71

Cf. A002586, A292559.

Select[Range[200], And[! IntegerQ @ Log2 @ #, Mod[FactorInteger[2^# + 1][[1, 1]], #] == 1] &] (* Michael De Vlieger, Sep 24 2017 *)
fQ[n_] := If[ OddQ@ n || IntegerQ@ Log2@ n || PrimeQ[2^n +1], False, Block[{p = 3}, While[PowerMod[2, n, p] +1 != p, p = NextPrime@ p]; Mod[p, n] == 1]] (* Robert G. Wilson v, Jan 01 2018 *)

isok(n) = my(e = valuation(n, 2)); (2^e != n) && ((vecmin(factor(2^n+1)[,1]) % n) == 1); \\ Michel Marcus, Nov 13 2017

a(9)-a(15) from Robert G. Wilson v, Jan 01 2018

a(16)-a(49) from Robert G. Wilson v, Jan 02 2018

cp's OEIS Frontend

A054992 Number of prime factors of 2^n + 1 (counted with multiplicity).

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A002587 Largest prime factor of 2^n + 1.

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A066135 a(n) = least number m > 1 such that sigma_n(m) = k*m for some k.

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A366719 Smallest prime dividing 12^n + 1.

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A366609 Smallest prime dividing 4^n + 1.

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A366670 Smallest prime dividing 6^n + 1.

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A066284 a(n) = A066135(4*n).

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A366671 Smallest prime dividing 8^n + 1.

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