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

Author

Gus Wiseman, Sep 03 2020

Keywords

• nonn

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.

Examples

			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)
		

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.

Programs

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}]

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).