cp's OEIS Frontend

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

%I A318810 #7 Dec 08 2018 20:47:17
%S A318810 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,
%T A318810 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,
%U A318810 1,54,120,10,21,1,8,5,35,1,180,1,1,70,9,14,28,1
%N A318810 Number of necklace permutations of a multiset whose multiplicities are the prime indices of n > 1.
%C A318810 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}.
%C A318810 A necklace is a finite sequence that is minimal among its cyclic permutations.
%C A318810 a(1) = 1 by convention.
%H A318810 Andrew Howroyd, <a href="/A318810/b318810.txt">Table of n, a(n) for n = 1..1000</a>
%F A318810 a(p) = 1 for prime p. - _Andrew Howroyd_, Dec 08 2018
%e A318810 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.
%t A318810 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A318810 neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
%t A318810 Table[Length[Select[Permutations[nrmptn[n]],neckQ]],{n,2,100}]
%o A318810 (PARI) sig(n)={my(f=factor(n)); concat(vector(#f~, i, vector(f[i,2], j, primepi(f[i,1]))))}
%o A318810 count(sig)={my(n=vecsum(sig)); sumdiv(gcd(sig), d, eulerphi(d)*(n/d)!/prod(i=1, #sig, (sig[i]/d)!))/n}
%o A318810 a(n)={if(n==1, 1, count(sig(n)))} \\ _Andrew Howroyd_, Dec 08 2018
%Y A318810 Cf. A000670, A008480, A019536, A056239, A060223, A112798, A298941, A305936, A318762, A318808, A318809.
%K A318810 nonn
%O A318810 1,8
%A A318810 _Gus Wiseman_, Sep 04 2018
%E A318810 a(1) inserted by _Andrew Howroyd_, Dec 08 2018