cp's OEIS Frontend

A381634 Number of multisets that can be obtained by taking the sum of each block of a set multipartition (multiset of sets) of the prime indices of n with distinct block-sums.

%I A381634 #11 Mar 09 2025 20:25:59
%S A381634 1,1,1,0,1,2,1,0,0,2,1,1,1,2,2,0,1,1,1,1,2,2,1,0,0,2,0,1,1,4,1,0,2,2,
%T A381634 2,1,1,2,2,0,1,5,1,1,1,2,1,0,0,1,2,1,1,0,2,0,2,2,1,3,1,2,1,0,2,5,1,1,
%U A381634 2,4,1,0,1,2,1,1,2,5,1,0,0,2,1,4,2,2,2
%N A381634 Number of multisets that can be obtained by taking the sum of each block of a set multipartition (multiset of sets) of the prime indices of n with distinct block-sums.
%C A381634 First differs from A050326 at a(30) = 4, A050326(30) = 5.
%C A381634 First differs from A339742 at a(42) = 5, A339742(42) = 4.
%C A381634 First differs from A381441 at a(30) = 4, A381441(30) = 5.
%C A381634 First differs from A381633 at a(210) = 10, A381633(210) = 12.
%C A381634 Also the number of multisets that can be obtained by taking the sums of prime indices of each factor in a factorization of n into squarefree numbers > 1 with distinct sums of prime indices (A056239).
%C A381634 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.
%C A381634 A multiset partition con be regarded as an arrow in the ranked poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).
%C A381634 Set multipartitions with distinct block-sums are generally not transitive. For example, we have arrows: {{1},{1,2}}: {1,1,2} -> {1,3} and {{1,3}}: {1,3} -> {4}, but there is no arrow {1,1,2} -> {4}.
%e A381634 The prime indices of 120 are {1,1,2,3}, with 3 ways:
%e A381634   {{1},{1,2,3}}
%e A381634   {{1,2},{1,3}}
%e A381634   {{1},{2},{1,3}}
%e A381634 with block-sums: {1,6}, {3,4}, {1,2,4}, so a(120) = 3.
%e A381634 The prime indices of 210 are {1,2,3,4}, with 12 ways:
%e A381634   {{1,2,3,4}}
%e A381634   {{1},{2,3,4}}
%e A381634   {{2},{1,3,4}}
%e A381634   {{3},{1,2,4}}
%e A381634   {{4},{1,2,3}}
%e A381634   {{1,2},{3,4}}
%e A381634   {{1,3},{2,4}}
%e A381634   {{1},{2},{3,4}}
%e A381634   {{1},{3},{2,4}}
%e A381634   {{1},{4},{2,3}}
%e A381634   {{2},{3},{1,4}}
%e A381634   {{1},{2},{3},{4}}
%e A381634 with block-sums: {10}, {1,9}, {2,8}, {3,7}, {4,6}, {3,7}, {4,6}, {1,2,7}, {1,3,6}, {1,4,5}, {2,3,5}, {1,2,3,4}, of which 10 are distinct, so a(210) = 10.
%t A381634 hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
%t A381634 sfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[sfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
%t A381634 Table[Length[Union[Sort[hwt/@#]&/@Select[sfacs[n],UnsameQ@@hwt/@#&]]],{n,100}]
%Y A381634 Without distinct block-sums we have A381078 (lower A381454), before sums A050320.
%Y A381634 For distinct blocks instead of sums we have A381441, before sums A050326, see A358914.
%Y A381634 Before taking sums we had A381633.
%Y A381634 Positions of 0 are A381806.
%Y A381634 Positions of 1 are A381870, superset of A293511.
%Y A381634 More on set multipartitions with distinct sums: A279785, A381717, A381718.
%Y A381634 A001055 counts multiset partitions, see A317141 (upper), A300383 (lower).
%Y A381634 A003963 gives product of prime indices.
%Y A381634 A055396 gives least prime index, greatest A061395.
%Y A381634 A056239 adds up prime indices, row sums of A112798.
%Y A381634 A265947 counts refinement-ordered pairs of integer partitions.
%Y A381634 Cf. A000720, A001222, A002846, A005117, A116540, A213242, A213385, A213427, A299202, A300385, A317142, A317143, A318360.
%K A381634 nonn
%O A381634 1,6
%A A381634 _Gus Wiseman_, Mar 06 2025