A335407 Number of anti-run permutations of the prime indices of n!.

1, 1, 1, 2, 0, 2, 3, 54, 0, 30, 105, 6090, 1512, 133056, 816480, 127209600, 0, 10090080, 562161600, 69864795000, 49989139200, 29593652088000, 382147120555200, 41810689605484800, 4359985823793600, 3025062801079038720, 49052072750637116160, 25835971971637227375360

Offset

0,4

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.

Conjecture: Only vanishes at n = 4 and n = 8.

a(16) = 0. Proof: 16! = 2^15 * m where bigomega(m) = A001222(m) = 13. We can't separate 15 1's with 13 other numbers. - David A. Corneth, Jul 04 2020

Links

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

Formula

a(n) = A335452(A000142(n)). - Andrew Howroyd, Feb 03 2021

Example

The a(0) = 1 through a(6) = 3 anti-run permutations:
  ()  ()  (1)  (1,2)  .  (1,2,1,3,1)  (1,2,1,2,1,3,1)
               (2,1)     (1,3,1,2,1)  (1,2,1,3,1,2,1)
                                      (1,3,1,2,1,2,1)

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)={count(factor(n!)[, 2])} \\ Andrew Howroyd, Feb 03 2021

Crossrefs

The version for Mersenne numbers is A335432.

Anti-run compositions are A003242.

Anti-run patterns are counted by 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.

Factorial numbers: A000142, A001222, A002982, A007489, A022559, A027423, A054991, A108731, A181819, A181821, A325272, A325273, A325617.

Sequence in context: A220222 A089839 A206823 * A151668 A086151 A099040

Adjacent sequences: A335404 A335405 A335406 * A335408 A335409 A335410

Keyword

nonn

Author

Gus Wiseman, Jul 01 2020

Extensions

Terms a(14) and beyond from Andrew Howroyd, Feb 03 2021

Status

approved