A336107 Number of permutations of the prime indices of n with at least one non-singleton run, or non-separations.

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 4, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 5, 1, 2, 0, 2, 0, 4, 0, 4, 0, 0, 0, 6, 0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 0, 9, 0, 0, 2, 2, 0, 0, 0, 5, 1, 0, 0, 6, 0, 0, 0

Offset

1,12

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.

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

Links

Table of n, a(n) for n=1..87.

Formula

a(n) = A008480(n) - A335452(n).

a(A000961(n)) = 0 if n is in A027883, otherwise 1.

a(A005117(n)) = 0.

a(n!) = A335459(n).

a(A006939(n)) = A022915(n).

Example

The a(n) non-separations for n = 12, 36, 60, 72, 180, 420:
  (11)  (112)  (1122)  (1123)  (11122)  (11223)  (11234)
        (211)  (1221)  (1132)  (11212)  (11232)  (11243)
               (2112)  (2113)  (11221)  (11322)  (11324)
               (2211)  (2311)  (12112)  (12213)  (11342)
                       (3112)  (12211)  (12231)  (11423)
                       (3211)  (21112)  (13122)  (11432)
                               (21121)  (13221)  (21134)
                               (21211)  (21123)  (21143)
                               (22111)  (21132)  (23114)
                                        (22113)  (23411)
                                        (22131)  (24113)
                                        (22311)  (24311)
                                        (23112)  (31124)
                                        (23211)  (31142)
                                        (31122)  (32114)
                                        (31221)  (32411)
                                        (32112)  (34112)
                                        (32211)  (34211)
                                                 (41123)
                                                 (41132)
                                                 (42113)
                                                 (42311)
                                                 (43112)
                                                 (43211)

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}]

Crossrefs

A005117 lists positions of zeros, with complement A013929.

A008480 counts permutations of prime indices, ranked by A333221.

A003242 and A335452 count separations, ranked by A333489.

A325535 counts inseparable partitions, ranked by A335448.

A325534 counts separable partitions, ranked by A335433.

Cf. A056239, A112798, A261962, A335451, A335452, A335460, A335489.

Sequence in context: A079127 A056674 A349798 * A350251 A367783 A363808

Adjacent sequences: A336104 A336105 A336106 * A336108 A336109 A336110

Keyword

nonn

Author

Gus Wiseman, Sep 03 2020

Status

approved