cp's OEIS Frontend

A327540 Number of factorizations of A327534(n), the n-th number that is 1, prime, or whose prime indices are relatively prime, into numbers > 1 satisfying the same conditions.

%I A327540 #6 Sep 18 2019 07:49:11
%S A327540 1,1,1,2,1,2,1,3,2,1,4,1,2,2,5,1,3,1,4,2,1,7,2,4,1,5,1,7,2,2,2,7,1,2,
%T A327540 7,1,4,1,4,3,2,1,12,3,2,4,1,4,2,7,2,1,11,1,2,11,5,1,4,2,5,1,13,1,2,3,
%U A327540 4,2,4,1,12,2,1,9,2,2,7,1,9,4,2,2,2,19,1
%N A327540 Number of factorizations of A327534(n), the n-th number that is 1, prime, or whose prime indices are relatively prime, into numbers > 1 satisfying the same conditions.
%C A327540 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. Numbers that are 1, prime, or whose prime indices are relatively prime are A327534. The number of divisors of n satisfying the same conditions is A327536(n).
%H A327540 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vSX9dPMGJhxB8rOknCGvOs6PiyhupdWNpqLsnphdgU6MEVqFBnWugAXidDhwHeKqZe_YnUqYeGOXsOk/pub">Sequences counting and encoding certain classes of multisets</a>
%e A327540 The a(74) = 9 factorizations of 84 together with the corresponding multiset partitions of {1,1,2,4}:
%e A327540   (2*2*3*7)  {{1},{1},{2},{4}}
%e A327540   (2*3*14)   {{1},{2},{1,4}}
%e A327540   (2*6*7)    {{1},{1,2},{4}}
%e A327540   (2*42)     {{1},{1,2,4}}
%e A327540   (3*4*7)    {{2},{1,1},{4}}
%e A327540   (3*28)     {{2},{1,1,4}}
%e A327540   (6*14)     {{1,2},{1,4}}
%e A327540   (7*12)     {{4},{1,1,2}}
%e A327540   (84)       {{1,1,2,4}}
%t A327540 nn=100;
%t A327540 facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
%t A327540 y=Select[Range[nn],#==1||PrimeQ[#]||GCD@@PrimePi/@First/@FactorInteger[#]==1&];
%t A327540 Table[Length[facsusing[Rest[y],n]],{n,y}]
%Y A327540 See link for additional cross-references.
%Y A327540 Cf. A006530, A056239, A112798, A281116, A289509, A327407.
%K A327540 nonn
%O A327540 1,4
%A A327540 _Gus Wiseman_, Sep 17 2019