A325611 -id:A325611 - 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

A325612 Width (number of leaves) of the rooted tree with Matula-Goebel number 2^n - 1.

Original entry on oeis.org

1, 1, 2, 2, 1, 4, 1, 4, 5, 3, 6, 7, 4, 5, 7, 6, 7, 11, 7, 7, 9, 10, 7, 13, 7, 11, 9, 11, 11, 13, 11, 12, 15, 16, 10, 19, 19, 15, 18, 16, 16, 18, 10, 18, 18, 17, 15, 21, 15, 18, 24, 23, 19, 23, 25, 25, 18, 26, 25, 28, 21, 21, 25, 23, 21, 29, 28, 31, 21, 24, 23

Offset: 1

Gus Wiseman, May 12 2019

nonn

Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example:
   11 = q(1) q(2) q(3) q(5)
   50 = q(1)^3 q(2)^2 q(3)^2
  360 = q(1)^6 q(2)^3 q(3)
For n > 1, a(n) is the multiplicity of q(1) = 2 in the q-factorization of 2^n - 1.

			The rooted tree with Matula-Goebel number 2047 = 2^11 - 1 is (((o)(o))(ooo(o))), which has 6 leaves (o's), so a(11) = 6.

Keith Briggs, Matula numbers and rooted trees.

Cf. A001222, A001221, A056239, A112798.
Matula-Goebel numbers: A007097, A061775, A109082, A109129, A196050, A317713.
Mersenne numbers: A046051, A046800, A059305, A325610, A325611, A325625.

mglv[n_]:=If[n==1,1,Total[Cases[FactorInteger[n],{p_,k_}:>mglv[PrimePi[p]]*k]]];
Table[mglv[2^n-1],{n,30}]

More terms from Jinyuan Wang, Feb 25 2025

A325610 Adjusted frequency depth of 2^n - 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 12 2019

nonn

The adjusted frequency depth of a positive integer n is 0 if n = 1, and otherwise it is 1 plus the number of times one must apply A181819 to reach a prime number, where A181819(k = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of k. For example, 180 has adjusted frequency depth 5 because we have: 180 -> 18 -> 6 -> 4 -> 3.

Cf. A001222, A001221, A056239, A071625, A112798, A323014, A325280.

Mersenne numbers: A046051, A046800, A059305, A325611, A325612, A325625.

fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#1]]&,ptn,Length[#1]>1&]]];
Table[fdadj[2^n-1],{n,100}]

A325625 Sorted prime signature of 2^n - 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, May 13 2019

The sorted prime signature of n is row n of A124010.

			We have 2^126 - 1 = 3^3 * 7^2 * 19 * 43 * 73 * 127 * 337 * 5419 * 92737 * 649657 * 77158673929, so row n = 126 is {1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3}.
Triangle begins:
  1
  1
  1
  1 1
  1
  1 2
  1
  1 1 1
  1 1
  1 1 1
  1 1
  1 1 1 2
  1
  1 1 1
  1 1 1
  1 1 1 1
  1
  1 1 1 3
  1
  1 1 1 1 2

Cf. A001222, A056239, A067255, A071625, A112798, A118914, A124010.

Mersenne numbers: A046051, A046800, A059305, A325610, A325611, A325612.

Table[Sort[Last/@FactorInteger[2^n-1]],{n,30}]

A335432 Number of anti-run permutations of the prime indices of Mersenne numbers A000225(n) = 2^n - 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 6, 2, 6, 2, 36, 1, 6, 6, 24, 1, 24, 1, 240, 6, 24, 2, 1800, 6, 6, 6, 720, 6, 1800, 1, 120, 24, 6, 24, 282240, 2, 6, 24, 15120, 2, 5760, 6, 5040, 720, 24, 6, 1451520, 2, 5040, 120, 5040, 6, 1800, 720, 40320, 24, 720, 2, 1117670400, 1, 6, 1800, 5040, 6

Offset: 1

Gus Wiseman, Jul 02 2020

nonn

An anti-run is a sequence with no adjacent equal parts.

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(1) = 1 through a(10) = 6 permutations:
  ()  (2)  (4)  (2,3)  (11)  (2,4,2)  (31)  (2,3,7)  (21,4)  (11,2,5)
                (3,2)                       (2,7,3)  (4,21)  (11,5,2)
                                            (3,2,7)          (2,11,5)
                                            (3,7,2)          (2,5,11)
                                            (7,2,3)          (5,11,2)
                                            (7,3,2)          (5,2,11)

Andrew Howroyd, Table of n, a(n) for n = 1..200

The version for factorial numbers is A335407.
Anti-run compositions are A003242.
Anti-run patterns are A005649.
Permutations of prime indices are A008480.
Anti-runs are ranked by A333489.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Strict permutations of prime indices are A335489.
Cf. A056239, A106351, A112798, A114938, A278990, A292884, A325535, A335125.
Mersenne numbers: A000225, A046051, A046800, A046801, A049093, A059305, A159611, 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,0,30}]

\\ See A335452 for count.
a(n) = {count(factor(2^n-1)[,2])} \\ Andrew Howroyd, Feb 03 2021

a(n) = A335452(A000225(n)).

Terms a(51) and beyond from Andrew Howroyd, Feb 03 2021

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).

A336105 Number of permutations of the prime indices of 2^n - 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 6, 2, 6, 2, 60, 1, 6, 6, 24, 1, 120, 1, 360, 12, 24, 2, 2520, 6, 6, 6, 720, 6, 2520, 1, 120, 24, 6, 24, 604800, 2, 6, 24, 20160, 2, 10080, 6, 5040, 720, 24, 6, 1814400, 2, 5040, 120, 5040, 6, 15120, 720, 40320, 24, 720, 2

Offset: 1

Gus Wiseman, Sep 03 2020

nonn

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(n) permutations for n = 2, 4, 6, 8, 21:
  (2)  (2,3)  (2,2,4)  (2,3,7)  (31,4,4,68)
       (3,2)  (2,4,2)  (2,7,3)  (31,4,68,4)
              (4,2,2)  (3,2,7)  (31,68,4,4)
                       (3,7,2)  (4,31,4,68)
                       (7,2,3)  (4,31,68,4)
                       (7,3,2)  (4,4,31,68)
                                (4,4,68,31)
                                (4,68,31,4)
                                (4,68,4,31)
                                (68,31,4,4)
                                (68,4,31,4)
                                (68,4,4,31)

A008480 is not restricted to predecessors of powers of 2.
A325617 is the version for factorial numbers.
A335489 counts strict permutations of prime indices.
Cf. A056239, A112798, A335432, A336104.
The numbers 2^n - 1: A000225, 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[Permutations[primeMS[2^n-1]]],{n,30}]

a(n) = A008480(2^n - 1).

a(n) = A336104(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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A325612 Width (number of leaves) of the rooted tree with Matula-Goebel number 2^n - 1.

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A325610 Adjusted frequency depth of 2^n - 1.

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A325625 Sorted prime signature of 2^n - 1.

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A335432 Number of anti-run permutations of the prime indices of Mersenne numbers A000225(n) = 2^n - 1.

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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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A336105 Number of permutations of the prime indices of 2^n - 1.

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