Seppo Mustonen - cp's OEIS frontend

A182590 Number of distinct prime factors of 2^n - 1 of the form k*n + 1.

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 1, 2, 3, 2, 1, 2, 2, 1, 2, 3, 3, 1, 2, 1, 2, 2, 3, 2, 2, 3, 2, 2, 4, 3, 3, 2, 3, 3, 3, 1, 4, 4, 3, 3, 2, 3, 2, 3, 5, 2, 2, 1, 2, 3, 4, 2, 3, 2, 3, 1, 4, 3, 3, 3, 4, 5, 3, 1, 5, 3, 2, 3, 4, 2, 3, 2, 4, 3

Offset: 2

Seppo Mustonen, Nov 22 2010

nonn

From Thomas Ordowski, Sep 08 2017: (Start)
By Bang's theorem, a(n) > 0 for all n > 1, see A186522.
Primes p such that a(p) = 1 are the Mersenne exponents A000043.
Composite numbers m for which a(m) = 1 are A292079.
a(n) >= A086251(n), where equality is for all prime numbers and for some composite numbers (among others for all odd prime powers p^k with k > 1).
Theorem: if n is prime, then a(n) = A046800(n).
Conjecture: if a(n) = A046800(n), then n is prime.
Problem: is a(n) < A046800(n) for every composite n? (End)

			For n=10 the prime factors of 2^n - 1 = 1023 are 3, 11 and 31, and 11 = n+1, 31 = 3n+1. Thus a(10)=2.

Charles R Greathouse IV, Table of n, a(n) for n = 2..1200 (terms 2..200 from Seppo Mustonen, terms 201..786 from Michel Marcus)
S. Mustonen, On prime factors of numbers m^n+-1, 2010.
Seppo Mustonen, On prime factors of numbers m^n+-1 [Local copy]

Cf. A046800, A086251, A186522.

m = 2; n = 2; nmax = 200;
While[n <= nmax, {l = FactorInteger[m^n - 1]; s = 0;
     For[i = 1, i <= Length[l],
      i++, {p = l[[i, 1]];
       If[IntegerQ[(p - 1)/n] == True, s = s + l[[i, 2]]];}];
     a[n] = s;} n++;];
Table[a[n], {n, 2, nmax}]

a(n) = my(f = factor(2^n-1)); sum(k=1, #f~, ((f[k,1]-1) % n)==0); \\ Michel Marcus, Sep 10 2017

Name edited by Thomas Ordowski, Sep 19 2017

A182594 Number of prime factors of form c*n+1 for numbers 7^n-1, counted with multiplicity.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 3, 3, 2, 2, 1, 4, 2, 3, 2, 4, 2, 3, 1, 4, 3, 4, 3, 3, 4, 4, 3, 3, 3, 2, 4, 3, 3, 3, 3, 4, 4, 5, 3, 4, 2, 4, 2, 4, 2, 3, 4, 4, 5, 5, 3, 5, 1, 6, 3, 4, 4, 5, 4, 6, 2, 3, 6, 6, 4, 6, 3, 8, 2, 5, 5, 5, 3, 2, 3, 7, 2, 5, 6, 7, 7, 3, 3, 9, 5, 4, 3, 6, 5, 5, 4, 3, 3, 5, 3, 11, 4, 6

Offset: 2

Seppo Mustonen, Nov 22 2010

nonn

			For n=9, 7^n-1 = 40353606 = 2*3^3*19*37*1063 has three prime factors of form, namely 19 = 2n+1, 37 = 4n+1, 1063 = 118n+1. Thus a(9) = 3.

Seppo Mustonen, On prime factors of numbers m^n+-1
Seppo Mustonen, On prime factors of numbers m^n+-1 [Local copy]

f:= proc(n) local F;
  F:= select(t -> t[1] mod n = 1, ifactors(7^n-1)[2]);
  convert(F[..,2],`+`)
end proc:
map(f, [$2..100]); # Robert Israel, Apr 29 2025

m = 7; n = 2; nmax = 80;
While[n <= nmax, {l = FactorInteger[m^n - 1]; s = 0;
     For[i = 1, i <= Length[l],
      i++, {p = l[[i, 1]];
       If[IntegerQ[(p - 1)/n] == True, s = s + l[[i, 2]]];}];
     a[n] = s;} n++;];
Table[a[n], {n, 2, nmax}]

Definition clarified and more terms from Robert Israel, Apr 29 2025

A182591 Number of prime factors of form cn+1 for numbers 3^n-1.

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 1, 1, 3, 2, 2, 1, 2, 1, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 2, 4, 3, 3, 4, 3, 2, 1, 5, 2, 4, 2, 4, 4, 2, 3, 5, 2, 3, 3, 3, 4, 5, 5, 4, 2, 4, 3, 6, 3, 2, 5, 6, 2, 3, 2, 5, 2, 2, 4, 5, 3, 3, 2, 3, 1, 4, 4, 5, 3, 5, 4, 9, 3, 3, 3, 5, 4, 5, 4, 3, 4

Offset: 2

Seppo Mustonen, Nov 22 2010

nonn

			For n=6, 3^n-1=728 has two prime factors of the form cn+1, namely 7=n+1 and 13=2n+1. Thus a(6)=2.

Seppo Mustonen, Table of n, a(n) for n = 2..170
S. Mustonen, On prime factors of numbers m^n+-1
Seppo Mustonen, On prime factors of numbers m^n+-1 [Local copy]

m = 3; n = 2; nmax = 170;
While[n <= nmax, {l = FactorInteger[m^n - 1]; s = 0;
     For[i = 1, i <= Length[l],
      i++, {p = l[[i, 1]];
       If[IntegerQ[(p - 1)/n] == True, s = s + l[[i, 2]]];}];
     a[n] = s;} n++;];
Table[a[n], {n, 2, nmax}]

A182592 Number of prime factors of form cn+1 for numbers 5^n-1.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 2, 3, 1, 2, 1, 3, 3, 2, 2, 3, 3, 3, 3, 4, 2, 3, 4, 3, 4, 3, 3, 5, 2, 3, 3, 4, 6, 3, 3, 6, 3, 5, 2, 6, 2, 3, 4, 4, 1, 2, 1, 6, 5, 3, 3, 7, 5, 3, 2, 5, 2, 7, 3, 5, 6, 4, 4, 7, 5, 8, 6, 8, 2, 3, 3, 6, 5, 5, 3, 7, 3, 4, 2, 6, 3, 3, 3, 6, 4, 4, 6, 5, 3, 2, 5, 4, 7, 5, 3, 4, 5, 7, 3, 10, 4, 5, 8, 6, 5, 2, 4, 7, 3, 6, 8, 5, 10, 2, 3, 6, 5, 7

Offset: 2

Seppo Mustonen, Nov 22 2010

nonn

			For n=10, 5^n-1=9765624=2^3*3*11*71*521 has three prime factors of the form cn+1, namely 11=n+1, 71=7n+1, 521=52n+1. Thus a(10)=3.

S. Mustonen, On prime factors of numbers m^n+-1
Seppo Mustonen, On prime factors of numbers m^n+-1 [Local copy]

m = 5; n = 2; nmax = 120;
While[n <= nmax, {l = FactorInteger[m^n - 1]; s = 0;
     For[i = 1, i <= Length[l],
      i++, {p = l[[i, 1]];
       If[IntegerQ[(p - 1)/n] == True, s = s + l[[i, 2]]];}];
     a[n] = s;} n++;];
Table[a[n], {n, 2, nmax}]
Table[Count[FactorInteger[5^n-1][[All,1]],?(Mod[#,n]==1&)],{n,2,130}] (* _Harvey P. Dale, Dec 11 2016 *)

A182593 Number of prime factors of form c*n+1 for numbers 6^n-1.

Original entry on oeis.org

2, 1, 2, 1, 3, 1, 1, 2, 3, 2, 3, 2, 3, 2, 3, 4, 3, 2, 3, 3, 3, 4, 3, 2, 5, 2, 4, 1, 4, 2, 3, 2, 6, 3, 5, 5, 4, 4, 3, 2, 4, 4, 4, 4, 6, 3, 5, 3, 4, 5, 6, 3, 5, 2, 5, 3, 4, 3, 7, 3, 3, 4, 4, 5, 6, 2, 4, 4, 8, 1, 7, 4, 8, 5, 4, 2, 9, 3, 5, 4, 5, 7, 4, 3, 5, 5, 4, 3, 6, 2, 6, 5, 4, 7, 8, 5, 6, 6, 7, 2, 11, 4, 7, 6, 7, 3, 6, 2, 6, 5, 6, 4, 6, 7, 4, 4, 4, 6, 6

Offset: 2

Seppo Mustonen, Nov 22 2010

nonn

			For n=6, 6^n-1=46655=5*7*31*43 and has three prime factors of form c*n+1, namely 7=n+1, 31=5*n+1, and 43=7*n+1. Thus a(6)=3. [Corrected by _N. J. A. Sloane_, Nov 16 2024]

Seppo Mustonen, On prime factors of numbers m^n+-1
Seppo Mustonen, On prime factors of numbers m^n+-1 [Local copy]

m = 6; n = 2; nmax = 120;
While[n <= nmax, {l = FactorInteger[m^n - 1]; s = 0;
     For[i = 1, i <= Length[l],
      i++, {p = l[[i, 1]];
       If[IntegerQ[(p - 1)/n] == True, s = s + l[[i, 2]]];}];
     a[n] = s;} n++;];
Table[a[n], {n, 2, nmax}]

A182595 Number of prime factors of form cn+1 for numbers 2^n+1.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 3, 1, 1, 2, 2, 1, 2, 2, 3, 2, 2, 2, 3, 2, 1, 2, 2, 1, 2, 1, 4, 2, 2, 1, 3, 3, 2, 2, 3, 2, 2, 3, 2, 3, 3, 1, 4, 2, 2, 4, 4, 2, 2, 2, 2, 2, 2, 2, 4, 2, 3, 3, 5, 1, 2, 3, 4, 5, 3, 2, 4, 2

Offset: 2

Seppo Mustonen, Nov 24 2010

nonn

Repeated prime factors are counted.

			For n=14, 2^n+1=16385=5*29*113 has two prime factors of form, namely 29=2n+1, 113=8n+1. Thus a(14)=2.

Seppo Mustonen, Table of n, a(n) for n = 2..250
S. Mustonen, On prime factors of numbers m^n+-1
Seppo Mustonen, On prime factors of numbers m^n+-1 [Local copy]

m = 2; n = 2; nmax = 250;
While[n <= nmax, {l = FactorInteger[m^n + 1]; s = 0;
     For[i = 1, i <= Length[l],
      i++, {p = l[[i, 1]];
       If[IntegerQ[(p - 1)/n] == True, s = s + l[[i, 2]]];}];
     a[n] = s;} n++;];
Table[a[n], {n, 2, nmax}]
Table[{p, e}=Transpose[FactorInteger[2^n+1]]; Sum[If[Mod[p[[i]], n] == 1, e[[i]], 0], {i, Length[p]}], {n, 2, 50}]

A182596 Number of prime factors of form cn+1 for numbers 3^n+1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 3, 1, 3, 2, 2, 2, 3, 2, 1, 4, 2, 2, 2, 2, 3, 1, 2, 1, 4, 2, 1, 2, 3, 2, 5, 2, 2, 2, 1, 3, 3, 2, 3, 2, 2, 3, 6, 2, 2, 2, 3, 3, 5, 2, 2, 5, 2, 3, 5, 1, 2, 1, 2, 3, 6, 3, 5, 3, 2, 3, 6, 4, 1, 2, 3, 4, 7, 3, 4, 5, 4, 5, 8, 3, 3, 3, 6, 2, 6, 2, 4, 4, 3, 5, 6, 3, 2, 5, 3, 4, 6, 4, 3, 7, 4, 4, 7, 7, 3, 4, 3, 3, 6, 1, 2, 5, 4, 4, 6, 2, 3, 4, 4, 5, 6, 3

Offset: 2

Seppo Mustonen, Nov 24 2010

nonn

Repeated prime factors are counted.

			For n=8, 3^n+1=6562=2*17*193 has two prime factors of form, namely 17=2n+1, 193=24n+1. Thus a(8)=2.

S. Mustonen, On prime factors of numbers m^n+-1
Seppo Mustonen, On prime factors of numbers m^n+-1 [Local copy]

m = 3; n = 2; nmax = 130;
While[n <= nmax, {l = FactorInteger[m^n + 1]; s = 0;
     For[i = 1, i <= Length[l],
      i++, {p = l[[i, 1]];
       If[IntegerQ[(p - 1)/n] == True, s = s + l[[i, 2]]];}];
     a[n] = s;} n++;];
Table[a[n], {n, 2, nmax}]
Table[{p, e}=Transpose[FactorInteger[3^n+1]]; Sum[If[Mod[p[[i]], n] == 1, e[[i]], 0], {i, Length[p]}], {n, 2, 50}]

A182597 Number of prime factors of form cn+1 for numbers 5^n+1.

Original entry on oeis.org

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

Offset: 2

Seppo Mustonen, Nov 24 2010

nonn

Repeated prime factors are counted.

			For n=11, 5^n+1=48828126=2*3*23*67*5281 has three prime factors of form, namely 23=2n+1, 67=6n+1, 5281=480n+1. Thus a(11)=3.

S. Mustonen, On prime factors of numbers m^n+-1
Seppo Mustonen, On prime factors of numbers m^n+-1 [Local copy]

m = 5; n = 2; nmax = 107;
While[n <= nmax, {l = FactorInteger[m^n + 1]; s = 0;
     For[i = 1, i <= Length[l],
      i++, {p = l[[i, 1]];
       If[IntegerQ[(p - 1)/n] == True, s = s + l[[i, 2]]]; }];
     a[n] = s; } n++; ];
Table[a[n], {n, 2, nmax}]
Table[{p,e}=Transpose[FactorInteger[5^n+1]]; Sum[If[Mod[p[[i]], n] == 1, e[[i]], 0], {i, Length[p]}], {n, 2, 50}]

A182598 Number of prime factors of form cn+1 for numbers 6^n+1.

Original entry on oeis.org

1, 2, 1, 2, 3, 2, 2, 1, 2, 1, 2, 3, 2, 2, 3, 2, 4, 2, 2, 1, 3, 2, 2, 2, 3, 3, 4, 3, 6, 1, 3, 4, 2, 5, 5, 3, 2, 5, 4, 3, 4, 1, 2, 2, 4, 1, 5, 3, 3, 6, 3, 4, 5, 4, 4, 3, 2, 1, 3, 2, 1, 3, 3, 3, 8, 4, 4, 2, 4, 3, 1, 5, 3, 5, 4, 1, 7, 5, 3, 3, 3, 4, 5, 3, 4, 7, 2, 2, 7, 5, 3, 2, 4, 5, 2, 3, 2, 4, 6

Offset: 2

Seppo Mustonen, Nov 24 2010

nonn

Repeated prime factors are counted.

			For n=6, 6^n-1=46655=5*7*31*43 has three prime factors of form, namely 7=n+1, 31=5n+1, 43=7n+1. Thus a(6)=3.

S. Mustonen, On prime factors of numbers m^n+-1
Seppo Mustonen, On prime factors of numbers m^n+-1 [Local copy]

m = 6; n = 2; nmax = 100;
While[n <= nmax, {l = FactorInteger[m^n + 1]; s = 0;
     For[i = 1, i <= Length[l],
      i++, {p = l[[i, 1]];
       If[IntegerQ[(p - 1)/n] == True, s = s + l[[i, 2]]];}];
     a[n] = s;} n++;];
Table[a[n], {n, 2, nmax}]
Table[{p, e}=Transpose[FactorInteger[6^n+1]]; Sum[If[Mod[p[[i]], n] == 1, e[[i]], 0], {i, Length[p]}], {n, 2, 50}]

A182599 Number of prime factors of form cn+1 for numbers 7^n+1.

Original entry on oeis.org

2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 4, 2, 1, 1, 2, 1, 2, 2, 3, 3, 3, 1, 1, 1, 2, 1, 4, 1, 4, 3, 3, 2, 3, 5, 4, 2, 1, 3, 3, 4, 2, 7, 3, 4, 4, 1, 3, 7, 4, 4, 3, 4, 3, 6, 5, 5, 4, 4, 3, 1, 3, 8, 3, 2, 5, 3, 3, 4, 4, 2, 5, 3, 1, 5, 5, 5, 4, 4, 3, 4, 3, 2, 5, 3, 3, 4, 2, 5, 4, 5, 4, 5, 3, 6, 6, 3, 5, 3, 3

Offset: 2

Seppo Mustonen, Nov 24 2010

nonn

Repeated prime factors are counted.

			For n=12, 7^12+1=13841287202=2*73*193*409*1201 has four prime factors of form, namely 73=6n+1, 193=16n+1, 409=34n+1, 1201=100n+1. Thus a(12)=4.

S. Mustonen, On prime factors of numbers m^n+-1
Seppo Mustonen, On prime factors of numbers m^n+-1 [Local copy]

m = 7; n = 2; nmax = 100;
While[n <= nmax, {l = FactorInteger[m^n + 1]; s = 0;
     For[i = 1, i <= Length[l],
      i++, {p = l[[i, 1]];
       If[IntegerQ[(p - 1)/n] == True, s = s + l[[i, 2]]];}];
     a[n] = s;} n++;];
Table[a[n], {n, 2, nmax}]
Table[{p,e}=Transpose[FactorInteger[7^n+1]]; Sum[If[Mod[p[[i]], n]==1, e[[i]], 0], {i, Length[p]}], {n, 2, 50}]

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User: Seppo Mustonen

A182590 Number of distinct prime factors of 2^n - 1 of the form k*n + 1.

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A182594 Number of prime factors of form c*n+1 for numbers 7^n-1, counted with multiplicity.

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A182591 Number of prime factors of form cn+1 for numbers 3^n-1.

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A182592 Number of prime factors of form cn+1 for numbers 5^n-1.

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A182593 Number of prime factors of form c*n+1 for numbers 6^n-1.

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A182595 Number of prime factors of form cn+1 for numbers 2^n+1.

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A182596 Number of prime factors of form cn+1 for numbers 3^n+1.

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A182597 Number of prime factors of form cn+1 for numbers 5^n+1.

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Examples

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Mathematica

A182598 Number of prime factors of form cn+1 for numbers 6^n+1.

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