cp's OEIS Frontend

A384179 Number of ways to choose strict integer partitions of each conjugate prime index of n.

%I A384179 #5 May 24 2025 11:00:31
%S A384179 1,1,1,1,1,1,1,2,1,1,1,2,1,1,1,2,1,2,1,2,1,1,1,2,1,1,4,2,1,2,1,3,1,1,
%T A384179 1,2,1,1,1,2,1,2,1,2,4,1,1,3,1,2,1,2,1,4,1,2,1,1,1,2,1,1,4,4,1,2,1,2,
%U A384179 1,2,1,3,1,1,4,2,1,2,1,3,4,1,1,2,1,1,1
%N A384179 Number of ways to choose strict integer partitions of each conjugate prime index of n.
%C A384179 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.
%e A384179 The prime indices of 180 are {1,1,2,2,3}, conjugate {5,3,1}, and we have choices:
%e A384179   {{5},{3},{1}}
%e A384179   {{5},{2,1},{1}}
%e A384179   {{4,1},{3},{1}}
%e A384179   {{4,1},{2,1},{1}}
%e A384179   {{3,2},{3},{1}}
%e A384179   {{3,2},{2,1},{1}}
%e A384179 so a(180) = 6.
%t A384179 fop[y_]:=Join@@@Tuples[strptns/@y];
%t A384179 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A384179 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A384179 Table[Length[fop[conj[prix[n]]]],{n,100}]
%Y A384179 Positions of 1 are A037143, complement A033942.
%Y A384179 For multiplicities instead of indices we have A050361.
%Y A384179 Adding up over all integer partitions gives A270995, disjoint A279790, strict A279375.
%Y A384179 The conjugate version is A357982, disjoint A383706.
%Y A384179 The disjoint case is A384005.
%Y A384179 A000041 counts integer partitions, strict A000009.
%Y A384179 A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
%Y A384179 A055396 gives least prime index, greatest A061395.
%Y A384179 A056239 adds up prime indices, row sums of A112798.
%Y A384179 A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
%Y A384179 A351293 counts non Look-and-Say or non section-sum partitions, ranks A351295 or A381433.
%Y A384179 Cf. A122111, A130091, A179009, A382913, A383707, A383710, A384010, A384011.
%K A384179 nonn
%O A384179 1,8
%A A384179 _Gus Wiseman_, May 23 2025