A335452 Number of separations (Carlitz compositions or anti-runs) of the prime indices of n.

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 6, 1, 0, 2, 2, 2, 2, 1, 2, 2, 0, 1, 6, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 6, 1, 2, 1, 0, 2, 6, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 2, 6, 1, 0, 0, 2, 1, 6, 2, 2, 2

Offset

1,6

Comments

The first term that is not a factorial number is a(180) = 12.

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.

A separation (or Carlitz composition) of a multiset is a permutation with no adjacent equal parts.

a(n) depends only on the prime signature of n. - Andrew Howroyd, Feb 03 2021

Links

Andrew Howroyd, Table of n, a(n) for n = 1..4096

Wikipedia, Permutation pattern

Example

The a(n) separations for n = 2, 6, 30, 180:
  (1)  (12)  (123)  (12123)
       (21)  (132)  (12132)
             (213)  (12312)
             (231)  (12321)
             (312)  (13212)
             (321)  (21213)
                    (21231)
                    (21312)
                    (21321)
                    (23121)
                    (31212)
                    (32121)

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, 100}]

Prog

(PARI)
F(i, j, r, t) = {sum(k=max(0, i-j), min(min(i, t), (i-j+t)\2), binomial(i, k)*binomial(r-i+1, t+i-j-2*k)*binomial(t-1, k-i+j))}
count(sig)={my(s=vecsum(sig), r=0, v=[1]); for(p=1, #sig, my(t=sig[p]); v=vector(s-r-t+1, j, sum(i=1, #v, v[i]*F(i-1, j-1, r, t))); r += t); v[1]}
a(n)={count(factor(n)[, 2])} \\ Andrew Howroyd, Feb 03 2021

Crossrefs

Separations are counted by A003242 and ranked by A333489.

Patterns are counted by A000670 and ranked by A333217.

Permutations of prime indices are counted by A008480.

Inseparable partitions are counted by A325535 and ranked by A335448.

Cf. A000961, A005117, A056239, A112798, A181796, A261962, A333221, A335451, A335454, A335465, A335489.

Sequence in context: A339742 A345164 A050326 * A355817 A056169 A286852

Adjacent sequences: A335449 A335450 A335451 * A335453 A335454 A335455

Keyword

nonn

Author

Gus Wiseman, Jun 21 2020

Status

approved