cp's OEIS Frontend

A325758 Irregular triangle read by rows giving the frequency span signature of n.

%I A325758 #6 May 19 2019 06:20:47
%S A325758 1,1,2,1,1,3,2,1,3,1,3,3,1,1,1,5,3,1,3,1,1,2,2,1,4,1,1,4,4,1,5,2,1,2,
%T A325758 2,1,3,1,1,1,6,2,1,1,2,3,1,1,3,1,5,2,1,1,4,1,2,1,5,1,2,2,1,3,1,1,2,1,
%U A325758 1,1,2,5,1,3,1,1,2,2,1,6,1,2,1,4,1,1,1
%N A325758 Irregular triangle read by rows giving the frequency span signature of n.
%C A325758 We define the frequency span of an integer partition to be the partition itself if it has no or only one block, and otherwise it is the multiset union of the partition and the frequency span of its multiplicities. For example, the frequency span of (3,2,2,1) is {1,2,2,3} U {1,1,2} U {1,2} U {1,1} U {2} = {1,1,1,1,1,1,2,2,2,2,2,3}. The frequency span of a positive integer is the frequency span of its prime indices (row n of A296150). Row n of this triangle gives an unsorted list of the multiplicities in the frequency span of n. For example, the frequency span of 30 is {1,1,1,1,2,3,3}, so row 30 is (4,1,2).
%e A325758 Triangle begins:
%e A325758   1
%e A325758   1
%e A325758   2 1
%e A325758   1
%e A325758   3 2
%e A325758   1
%e A325758   3 1
%e A325758   3
%e A325758   3 1 1
%e A325758   1
%e A325758   5 3
%e A325758   1
%e A325758   3 1 1
%e A325758   2 2 1
%e A325758   4 1
%e A325758   1
%e A325758   4 4
%e A325758   1
%e A325758   5 2 1
%e A325758   2 2 1
%e A325758   3 1 1
%e A325758   1
%e A325758   6 2 1
%e A325758   1 2
%e A325758   3 1 1
%e A325758   3 1
%e A325758   5 2 1
%e A325758   1
%e A325758   4 1 2
%t A325758 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A325758 freqspan[ptn_]:=If[Length[ptn]<=1,ptn,Sort[Join[ptn,freqspan[Sort[Length/@Split[ptn]]]]]];
%t A325758 Table[Length/@Split[freqspan[primeMS[n]]],{n,30}]
%Y A325758 Row sums are A325249.
%Y A325758 Row lengths are A325759.
%Y A325758 Run-lengths of A325757.
%Y A325758 Row n is the unsorted prime signature of A325760(n).
%Y A325758 Cf. A001221, A001222, A056239, A071625, A112798, A181819, A182857, A290822, A323014,A325277.
%K A325758 nonn,tabf
%O A325758 1,3
%A A325758 _Gus Wiseman_, May 19 2019