cp's OEIS Frontend

A382426 MM-numbers of sets of constant multisets with distinct sums.

%I A382426 #8 Apr 03 2025 14:57:53
%S A382426 1,2,3,5,6,7,10,11,14,15,17,19,21,22,23,30,31,33,34,38,41,42,46,51,53,
%T A382426 55,57,59,62,66,67,69,77,82,83,85,93,95,97,102,103,106,109,110,114,
%U A382426 115,118,119,123,127,131,133,134,138,154,155,157,159,161,165,166
%N A382426 MM-numbers of sets of constant multisets with distinct sums.
%C A382426 Also products of prime numbers of prime power index with distinct sums of prime indices.
%C A382426 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, sum A056239. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.
%F A382426 Equals A302492 /\ A326535.
%e A382426 The terms together with their prime indices of prime indices begin:
%e A382426    1: {}
%e A382426    2: {{}}
%e A382426    3: {{1}}
%e A382426    5: {{2}}
%e A382426    6: {{},{1}}
%e A382426    7: {{1,1}}
%e A382426   10: {{},{2}}
%e A382426   11: {{3}}
%e A382426   14: {{},{1,1}}
%e A382426   15: {{1},{2}}
%e A382426   17: {{4}}
%e A382426   19: {{1,1,1}}
%e A382426   21: {{1},{1,1}}
%e A382426   22: {{},{3}}
%e A382426   23: {{2,2}}
%e A382426   30: {{},{1},{2}}
%t A382426 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A382426 Select[Range[100],UnsameQ@@Total/@prix/@prix[#]&&And@@SameQ@@@prix/@prix[#]&]
%Y A382426 Twice-partitions of this type are counted by A279786.
%Y A382426 For just constant blocks we have A302492.
%Y A382426 For just distinct sums we have A326535.
%Y A382426 Factorizations of this type are counted by A381635.
%Y A382426 For strict instead of constant blocks we have A382201.
%Y A382426 Normal multiset partitions of this type are counted by A382203.
%Y A382426 For equal instead of distinct sums we have A382215.
%Y A382426 An opposite version is A382304.
%Y A382426 A055396 gives least prime index, greatest A061395.
%Y A382426 A056239 adds up prime indices, row sums of A112798.
%Y A382426 Cf. A000688, A000720, A000961, A302242, A302494, A321469, A326534, A355743, A356065, A381636, A381716.
%K A382426 nonn
%O A382426 1,2
%A A382426 _Gus Wiseman_, Apr 01 2025