A335459 Number of permutations of the prime indices of n! with at least one non-singleton run.

0, 0, 0, 0, 4, 18, 102, 786, 3960, 51450, 675570, 10804710, 139674024, 2793377664, 58662908640, 1798893694080, 26985313555200, 782574083010720, 25992638958686400, 857757034323189000, 30021498596590300800, 1563341714743040232000, 64179292280096037844800, 2631350957341279888915200

Offset

0,5

Comments

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.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

Formula

A008480(n!) = a(n) + A335407(n).

Example

The a(4) = 4 and a(5) = 18 permutations:
  (1,1,1,2)  (1,1,1,2,3)
  (1,1,2,1)  (1,1,1,3,2)
  (1,2,1,1)  (1,1,2,1,3)
  (2,1,1,1)  (1,1,2,3,1)
             (1,1,3,1,2)
             (1,1,3,2,1)
             (1,2,1,1,3)
             (1,2,3,1,1)
             (1,3,1,1,2)
             (1,3,2,1,1)
             (2,1,1,1,3)
             (2,1,1,3,1)
             (2,1,3,1,1)
             (2,3,1,1,1)
             (3,1,1,1,2)
             (3,1,1,2,1)
             (3,1,2,1,1)
             (3,2,1,1,1)

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[Select[Permutations[primeMS[n!]], MatchQ[#, {___, x_, x_, ___}]&]], {n, 0, 10}]

Prog

(PARI) \\ See A335452 for count.
a(n)={my(sig=factor(n!)[, 2]); vecsum(sig)!/vecprod([k! | k<-sig]) - count(sig)} \\ Andrew Howroyd, Apr 17 2021

Crossrefs

The anti-run version is A335407.

Anti-runs are ranked by A333489.

Anti-run compositions are A003242.

Anti-run patterns are A005649.

Permutations of prime indices are A008480.

Permutations of prime indices of n! are A325617.

Anti-run permutations of prime indices are A335452.

Cf. A001222, A022559, A056239, A106351, A112798, A114938, A325535, A335125.

Factorial numbers: A000142, A002982, A007489, A027423, A054991, A108731, A325272, A325273, A325617.

Sequence in context: A064852 A229286 A191365 * A159666 A349021 A316186

Adjacent sequences: A335456 A335457 A335458 * A335460 A335461 A335462

Keyword

nonn

Author

Gus Wiseman, Jul 03 2020

Extensions

a(11)-a(13) from Vaclav Kotesovec, Jul 07 2020

Terms a(14) and beyond from Andrew Howroyd, Apr 17 2021

Status

approved