cp's OEIS Frontend

A381991 Numbers whose prime indices have a unique multiset partition into constant multisets with distinct sums.

%I A381991 #8 Mar 23 2025 08:40:34
%S A381991 1,2,3,4,5,6,7,9,10,11,13,14,15,17,18,19,20,21,22,23,24,25,26,28,29,
%T A381991 30,31,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,49,50,51,52,53,55,
%U A381991 57,58,59,61,62,65,66,67,68,69,70,71,73,74,75,76,77,78,79
%N A381991 Numbers whose prime indices have a unique multiset partition into constant multisets with distinct sums.
%C A381991 Also numbers with a unique factorization into prime powers with distinct sums of prime indices.
%C A381991 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.
%e A381991 The prime indices of 270 are {1,2,2,2,3}, and there are two multiset partitions into constant multisets with distinct sums: {{1},{2},{3},{2,2}} and {{1},{3},{2,2,2}}, so 270 is not in the sequence.
%e A381991 The prime indices of 300 are {1,1,2,3,3}, of which there are no multiset partitions into constant multisets with distinct sums, so 300 is not in the sequence.
%e A381991 The prime indices of 360 are {1,1,1,2,2,3}, of which there is only one multiset partition into constant multisets with distinct sums: {{1},{1,1},{3},{2,2}}, so 360 is in the sequence.
%e A381991 The terms together with their prime indices begin:
%e A381991     1: {}
%e A381991     2: {1}
%e A381991     3: {2}
%e A381991     4: {1,1}
%e A381991     5: {3}
%e A381991     6: {1,2}
%e A381991     7: {4}
%e A381991     9: {2,2}
%e A381991    10: {1,3}
%e A381991    11: {5}
%e A381991    13: {6}
%e A381991    14: {1,4}
%e A381991    15: {2,3}
%e A381991    17: {7}
%e A381991    18: {1,2,2}
%e A381991    19: {8}
%e A381991    20: {1,1,3}
%e A381991    21: {2,4}
%e A381991    22: {1,5}
%e A381991    23: {9}
%e A381991    24: {1,1,1,2}
%e A381991    25: {3,3}
%t A381991 hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
%t A381991 pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
%t A381991 Select[Range[100],Length[Select[pfacs[#],UnsameQ@@hwt/@#&]]==1&]
%Y A381991 For distinct blocks instead of block-sums we have A004709, counted by A000726.
%Y A381991 Twice-partitions of this type are counted by A279786.
%Y A381991 MM-numbers of these multiset partitions are A326535 /\ A355743.
%Y A381991 These are the positions of 1 in A381635.
%Y A381991 For no choices we have A381636 (zeros of A381635), counted by A381717.
%Y A381991 For strict instead of constant blocks we have A381870, counted by A382079.
%Y A381991 Partitions of this type (unique into constant with distinct) are counted by A382301.
%Y A381991 Normal multiset partitions of this type are counted by A382203.
%Y A381991 A001055 counts multiset partitions, see A317141 (upper), A300383 (lower), A265947.
%Y A381991 A055396 gives least prime index, greatest A061395.
%Y A381991 A056239 adds up prime indices, row sums of A112798.
%Y A381991 Cf. A000688, A000720, A001222, A003963, A005117, A047966, A050361, A293511, A300385, A381716.
%Y A381991 Cf. A006171, A279784, A295935.
%K A381991 nonn
%O A381991 1,2
%A A381991 _Gus Wiseman_, Mar 22 2025