A318810 Number of necklace permutations of a multiset whose multiplicities are the prime indices of n > 1.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 2, 6, 1, 6, 1, 4, 3, 1, 1, 12, 4, 1, 16, 5, 1, 10, 1, 24, 3, 1, 5, 30, 1, 1, 4, 20, 1, 15, 1, 6, 30, 1, 1, 60, 10, 20, 4, 7, 1, 90, 7, 30, 5, 1, 1, 60, 1, 1, 54, 120, 10, 21, 1, 8, 5, 35, 1, 180, 1, 1, 70, 9, 14, 28, 1

Offset

1,8

Comments

This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

A necklace is a finite sequence that is minimal among its cyclic permutations.

a(1) = 1 by convention.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Formula

a(p) = 1 for prime p. - Andrew Howroyd, Dec 08 2018

Example

The a(21) = 3 necklace permutations of {1,1,1,1,2,2} are: (111122), (111212), (112112). Only the first two are Lyndon words, the third being periodic.

Mathematica

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
neckQ[q_]:=Array[OrderedQ[{q, RotateRight[q, #]}]&, Length[q]-1, 1, And];
Table[Length[Select[Permutations[nrmptn[n]], neckQ]], {n, 2, 100}]

Prog

(PARI) sig(n)={my(f=factor(n)); concat(vector(#f~, i, vector(f[i, 2], j, primepi(f[i, 1]))))}
count(sig)={my(n=vecsum(sig)); sumdiv(gcd(sig), d, eulerphi(d)*(n/d)!/prod(i=1, #sig, (sig[i]/d)!))/n}
a(n)={if(n==1, 1, count(sig(n)))} \\ Andrew Howroyd, Dec 08 2018

Crossrefs

Cf. A000670, A008480, A019536, A056239, A060223, A112798, A298941, A305936, A318762, A318808, A318809.

Sequence in context: A238280 A326644 A120964 * A319989 A338117 A353468

Adjacent sequences: A318807 A318808 A318809 * A318811 A318812 A318813

Keyword

nonn

Author

Gus Wiseman, Sep 04 2018

Extensions

a(1) inserted by Andrew Howroyd, Dec 08 2018

Status

approved