cp's OEIS Frontend

A325757 Irregular triangle read by rows giving the frequency span of n.

%I A325757 #9 May 19 2019 06:18:15
%S A325757 1,2,1,1,2,3,1,1,1,2,2,4,1,1,1,3,2,2,2,1,1,1,2,3,5,1,1,1,1,1,2,2,2,6,
%T A325757 1,1,1,2,4,1,1,2,2,3,1,1,1,1,4,7,1,1,1,1,2,2,2,2,8,1,1,1,1,1,2,2,3,1,
%U A325757 1,2,2,4,1,1,1,2,5,9,1,1,1,1,1,1,2,2,3
%N A325757 Irregular triangle read by rows giving the frequency span of n.
%C A325757 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).
%e A325757 Triangle begins:
%e A325757    1:
%e A325757    2: 1
%e A325757    3: 2
%e A325757    4: 1 1 2
%e A325757    5: 3
%e A325757    6: 1 1 1 2 2
%e A325757    7: 4
%e A325757    8: 1 1 1 3
%e A325757    9: 2 2 2
%e A325757   10: 1 1 1 2 3
%e A325757   11: 5
%e A325757   12: 1 1 1 1 1 2 2 2
%e A325757   13: 6
%e A325757   14: 1 1 1 2 4
%e A325757   15: 1 1 2 2 3
%e A325757   16: 1 1 1 1 4
%e A325757   17: 7
%e A325757   18: 1 1 1 1 2 2 2 2
%e A325757   19: 8
%e A325757   20: 1 1 1 1 1 2 2 3
%e A325757   21: 1 1 2 2 4
%e A325757   22: 1 1 1 2 5
%e A325757   23: 9
%e A325757   24: 1 1 1 1 1 1 2 2 3
%e A325757   25: 2 3 3
%e A325757   26: 1 1 1 2 6
%e A325757   27: 2 2 2 3
%e A325757   28: 1 1 1 1 1 2 2 4
%t A325757 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A325757 freqspan[ptn_]:=If[Length[ptn]<=1,ptn,Sort[Join[ptn,freqspan[Sort[Length/@Split[ptn]]]]]];
%t A325757 Table[freqspan[primeMS[n]],{n,15}]
%Y A325757 Row lengths are A325249.
%Y A325757 Run-lengths are A325758.
%Y A325757 Number of distinct terms in row n is A325759(n).
%Y A325757 Cf. A001221, A001222, A056239, A071625, A112798, A181819, A182857, A290822, A323014, A324843, A325277, A325755, A325760.
%K A325757 nonn,tabf
%O A325757 1,2
%A A325757 _Gus Wiseman_, May 19 2019