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

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

Author

Gus Wiseman, Jul 02 2020

Keywords

• nonn

Comments

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.

Examples

			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)
		

Links

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

Crossrefs

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.

Programs

Mathematica

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}]

PARI

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

Formula

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

Extensions

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