A001265 -id:A001265 - cp's OEIS frontend

A046051 Number of prime factors of Mersenne number M(n) = 2^n - 1 (counted with multiplicity).

0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 2, 5, 1, 3, 3, 4, 1, 6, 1, 6, 4, 4, 2, 7, 3, 3, 3, 6, 3, 7, 1, 5, 4, 3, 4, 10, 2, 3, 4, 8, 2, 8, 3, 7, 6, 4, 3, 10, 2, 7, 5, 7, 3, 9, 6, 8, 4, 6, 2, 13, 1, 3, 7, 7, 3, 9, 2, 7, 4, 9, 3, 14, 3, 5, 7, 7, 4, 8, 3, 10, 6, 5, 2, 14, 3, 5, 6, 10, 1, 13, 5, 9, 3, 6, 5, 13, 2, 5, 8

Offset: 1

Eric W. Weisstein

nonn

Length of row n of A001265.

			a(4) = 2 because 2^4 - 1 = 15 = 3*5.
From _Gus Wiseman_, Jul 04 2019: (Start)
The sequence of Mersenne numbers together with their prime indices begins:
        1: {}
        3: {2}
        7: {4}
       15: {2,3}
       31: {11}
       63: {2,2,4}
      127: {31}
      255: {2,3,7}
      511: {4,21}
     1023: {2,5,11}
     2047: {9,24}
     4095: {2,2,3,4,6}
     8191: {1028}
    16383: {2,14,31}
    32767: {4,11,36}
    65535: {2,3,7,55}
   131071: {12251}
   262143: {2,2,2,4,8,21}
   524287: {43390}
  1048575: {2,3,3,5,11,13}
(End)

Sean A. Irvine, Table of n, a(n) for n = 1..1206 (terms 1..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.
Alex Kontorovich, Jeff Lagarias, On Toric Orbits in the Affine Sieve, arXiv:1808.03235 [math.NT], 2018.
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018.
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Mersenne Number

bigomega(b^n-1): A057951 (b=10), A057952 (b=9), A057953 (b=8), A057954 (b=7), A057955 (b=6), A057956 (b=5), A057957 (b=4), A057958 (b=3), this sequence (b=2).
Cf. A000043, A000668, A001348, A054988, A054989, A054990, A054991, A054992, A085021.
Cf. A000225, A001221, A001222, A046800, A049093, A059305, A325610, A325611, A325612, A325625.

a[q_] := Module[{x, n}, x=FactorInteger[2^n-1]; n=Length[x]; Sum[Table[x[i][2], {i, n}][j], {j, n}]]
a[n_Integer] := PrimeOmega[2^n - 1]; Table[a[n], {n,200}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)

a(n)=bigomega(2^n-1) \\ Charles R Greathouse IV, Apr 01 2013

Mobius transform of A085021. - T. D. Noe, Jun 19 2003

a(n) = A001222(A000225(n)). - Michel Marcus, Jun 06 2019

A060443 Table T(n,k) in which n-th row lists prime factors of 2^n - 1 (n >= 2), without repetition.

Original entry on oeis.org

0, 1, 3, 7, 3, 5, 31, 3, 7, 127, 3, 5, 17, 7, 73, 3, 11, 31, 23, 89, 3, 5, 7, 13, 8191, 3, 43, 127, 7, 31, 151, 3, 5, 17, 257, 131071, 3, 7, 19, 73, 524287, 3, 5, 11, 31, 41, 7, 127, 337, 3, 23, 89, 683, 47, 178481, 3, 5, 7, 13, 17, 241

Offset: 0

N. J. A. Sloane

For n > 1, the length of row n is A046800(n). - T. D. Noe, Aug 06 2007

			From _Wolfdieter Lang_, Sep 23 2017: (Start)
The irregular triangle T(n,k) begins for n >= 2:
n\k      1   2    3    4   5
2:       3
3:       7
4:       3   5
5:      31
6:       3   7
7:     127
8:       3   5   17
9:       7  73
10:      3  11   31
11:     23  89
12:      3   5    7   13
13:   8191
14:      3  43  127
15:      7  31  151
16:      3   5   17  257
17: 131071
18:      3   7   19   73
19: 524287
20:      3   5   11   31  41
... (End)

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

T. D. Noe, Rows n=0..500 of triangle, flattened (derived from Brillhart et al.)
Joerg Arndt, Rows n=1..1200 of triangle when repetitions are included (derived from Brillhart et al.)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Jeroen Demeyer, Machine-readable Cunningham Tables [Broken link]
S. S. Wagstaff, Jr., The Cunningham Project
Chai Wah Wu, Tables from the Cunningham Project in machine-readable JSON format.

Cf. A001265, A064078.

Array[FactorInteger[2^# - 1][[All, 1]] &, 25, 0] (* Paolo Xausa, Apr 18 2024 *)

A292079 Composite numbers m such that 2^m - 1 has a single prime factor of the form k*m + 1.

Original entry on oeis.org

4, 6, 8, 9, 12, 20, 24, 27, 33, 49, 69, 77, 145, 425, 447, 567

Offset: 1

Michel Marcus, Sep 12 2017

From Thomas Ordowski, Sep 12 2017: (Start)
Composite numbers m such that A182590(m) = 1.
Problem: are there infinitely many such numbers?
Note that this single prime factor p is the only primitive prime factor of 2^m - 1 for all such m except 6, i.e., the multiplicative order of 2 modulo p is m. (End)
After 567, the only numbers < 1200 that may possibly be terms are 961, 1037, 1111, and 1115. - Jon E. Schoenfield, Dec 03 2017
a(17) > 1206. - Amiram Eldar, Apr 01 2021

Cf. A001265, A002326, A182590.

Select[Range@ 150, And[CompositeQ@ #, Function[{m, p}, Total@ Boole@ Map[Divisible[# - 1, m] &, p] == 1] @@ {#, FactorInteger[2^# - 1][[All, 1]]}] &] (* Michael De Vlieger, Dec 06 2017 *)

lista(nn) = forcomposite(n=1, nn, my(f = factor(2^n-1)); if (sum(k=1, #f~, ((f[k, 1]-1) % n)==0) == 1, print1(n, ", ")));

Erroneous terms 841 and 1127 and possible (but unconfirmed, and not necessarily next) term 1037 deleted by Jon E. Schoenfield, Dec 03 2017

A297294 Number of primitive Pythagorean triples (PPTs) that have 2^n-1 as the length of their odd leg where n is the sequence index.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 4, 2, 4, 2, 8, 1, 4, 4, 8, 1, 8, 1, 16, 4, 8, 2, 32, 4, 4, 4, 32, 4, 32, 1, 16, 8, 4, 8, 128, 2, 4, 8, 64, 2, 32, 4, 64, 32, 8, 4, 256, 2, 64, 16, 64, 4, 32, 32, 128, 8, 32, 2, 1024, 1, 4, 32, 64, 4, 128, 2, 64, 8, 256, 4, 2048, 4, 16, 64, 64, 8, 64, 4, 256, 32, 16, 2, 2048, 4, 16, 32, 512, 1, 1024

Offset: 1

Frank M Jackson, Jan 04 2018

nonn

2^n-1 for n = 0 and 1 give the Mersenne numbers 0 and 1, neither of which can be the side length of a PPT. For n > 1, all Mersenne numbers are congruent to 3 mod 4. Consequently, no Mersenne number can be the length of the hypotenuse of a PPT.
If 2^n-1 is the length of the odd leg of a PPT its divisors can provide a set of pairs {x, y} such that for each pair, x*y = 2^n-1, x < y and gcd(x, y) = 1. Using Euclid's parametric generators for PPTs (s^2+t^2, 2s*t, s^2-t^2) with s > t > 0 as positive integers, gcd(s, t) = 1 and s+t odd it is possible to generate all PPTs with 2^n-1 as the length of the odd leg providing that s = (x+y)/2 and t = (y-x)/2.
If 2^n-1 has d distinct prime factors (A046800(n)), then the set of pairs {x, y} such that x*y = 2^n-1, x < y and gcd(x, y) = 1 has a cardinality of 2^(d-1). This is because an integer m consisting of d distinct factors will have 2^d divisors and will generate pairs {x', y'} such that x'*y' = m, x' < y' and gcd(x', y') = 1 with a cardinality of 2^(d-1). Let m be the product of the distinct factor of 2^n-1 and r be the remainder consisting of the remaining repeated prime factors where m*r = 2^n-1. Then there has to be a 1 to 1 correspondence between the set of pairs {x', y'} created from the distinct prime factors of 2^n-1 and {x, y} created from all the prime factors of 2^n-1 whenever the repeated prime factors of r are combined with the distinct factors of m in the pairs {x, y} in order to preserve gcd(x, y) = 1.

			a(6)=2, because 2^6-1 = 63 gives pairs {1, 63}, {3, 21}, {7, 9} whose members when multiplied give 63. However, only two of these pairs are coprime and will generate PPTs.

Frank M Jackson, Table of n, a(n) for n = 1..928

Cf. A001265, A046800.

pairs[n_] := Module[{m=2^n-1, lst=Divisors[2^n-1]}, Table[{lst[[l]], m/lst[[l]]}, {l, 1, Length[lst]/2}]]; Table[Length@Select[pairs[n], GCD@@#==1 &], {n, 1, 100}]
a[n_] := If[n==1, 0, 2^(Length@FactorInteger[2^n-1]-1)]; Array[a, 100]

For n=1, a(n)=0 otherwise a(n)=2^(A046800(n)-1).

A336104 Number of permutations of the prime indices of A000225(n) = 2^n - 1 with at least one non-singleton run.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 96, 0, 120, 6, 0, 0, 720, 0, 0, 0, 0, 0, 720, 0, 0, 0, 0, 0, 322560, 0, 0, 0, 5040, 0, 4320, 0, 0, 0, 0, 0, 362880, 0, 0

Offset: 1

Gus Wiseman, Sep 03 2020

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(21) = 6 permutations of {4, 4, 31, 68}:
  (4,4,31,68)
  (4,4,68,31)
  (31,4,4,68)
  (31,68,4,4)
  (68,4,4,31)
  (68,31,4,4)

A335432 is the anti-run version.
A335459 is the version for factorial numbers.
A336105 counts all permutations of this multiset.
A336107 is not restricted to predecessors of powers of 2.
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A008480 counts permutations of prime indices.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A333489 ranks anti-run compositions.
A335433 lists numbers whose prime indices have an anti-run permutation.
A335448 lists numbers whose prime indices have no anti-run permutation.
A335452 counts anti-run permutations of prime indices.
A335489 counts strict permutations of prime indices.
Cf. A056239, A106351, A112798, A114938, A292884, A336102.
The numbers 2^n - 1: A000225, A001265, A001348, A046051, A046800, A046801, A049093, A325610, A325611, A325612, A325625.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[2^n-1]],MatchQ[#,{_,x_,x_,_}]&]],{n,30}]

a(n) = A336107(2^n - 1).

a(n) = A336105(n) - A335432(n).

cp's OEIS Frontend

A046051 Number of prime factors of Mersenne number M(n) = 2^n - 1 (counted with multiplicity).

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A060443 Table T(n,k) in which n-th row lists prime factors of 2^n - 1 (n >= 2), without repetition.

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A292079 Composite numbers m such that 2^m - 1 has a single prime factor of the form k*m + 1.

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A297294 Number of primitive Pythagorean triples (PPTs) that have 2^n-1 as the length of their odd leg where n is the sequence index.

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A336104 Number of permutations of the prime indices of A000225(n) = 2^n - 1 with at least one non-singleton run.

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