cp's OEIS Frontend

A381078 Number of multisets that can be obtained by partitioning the prime indices of n into a multiset of sets (set multipartition) and taking their sums.

%I A381078 #13 Apr 01 2025 12:16:12
%S A381078 1,1,1,1,1,2,1,1,1,2,1,2,1,2,2,1,1,2,1,2,2,2,1,2,1,2,1,2,1,5,1,1,2,2,
%T A381078 2,3,1,2,2,2,1,5,1,2,2,2,1,2,1,2,2,2,1,2,2,2,2,2,1,6,1,2,2,1,2,5,1,2,
%U A381078 2,5,1,3,1,2,2,2,2,5,1,2,1,2,1,6,2,2,2
%N A381078 Number of multisets that can be obtained by partitioning the prime indices of n into a multiset of sets (set multipartition) and taking their sums.
%C A381078 First differs from A050320 at a(210) = 13, A050320(210) = 15. This comes from the set multipartitions {{3},{1,2,4}} and {{1,2},{3,4}}, and from {{4},{1,2,3}} and {{1,3},{2,4}}.
%C A381078 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.
%C A381078 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 A381078 A multiset partition can 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 A381078 Set multipartitions 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 set multipartition {1,1,2} -> {4}.
%H A381078 Robert Price, <a href="/A381078/b381078.txt">Table of n, a(n) for n = 1..1000</a>
%F A381078 a(A002110(n)) = A066723(n).
%e A381078 The prime indices of 60 are {1,1,2,3}, with set multipartitions:
%e A381078   {{1},{1,2,3}}
%e A381078   {{1,2},{1,3}}
%e A381078   {{1},{1},{2,3}}
%e A381078   {{1},{2},{1,3}}
%e A381078   {{1},{3},{1,2}}
%e A381078   {{1},{1},{2},{3}}
%e A381078 with block-sums: {1,6}, {3,4}, {1,1,5}, {1,2,4}, {1,3,3}, {1,1,2,3}, which are all different multisets, so a(60) = 6.
%t A381078 hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
%t A381078 sqfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[sqfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
%t A381078 Table[Length[Union[Sort[hwt/@#]&/@sqfacs[n]]],{n,100}]
%Y A381078 Before taking sums we had A050320, strict A050326 (zeros A293243), distinct sums A381633.
%Y A381078 For distinct blocks we have A381441.
%Y A381078 The lower version is A381454.
%Y A381078 For distinct block-sums we have A381634.
%Y A381078 Other multiset partitions of prime indices:
%Y A381078 - For multisets of constant multisets (A000688) see A381455 (upper), A381453 (lower).
%Y A381078 - For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
%Y A381078 - For strict multiset partitions (A045778) see A381452.
%Y A381078 - For sets of constant multisets (A050361) see A381717.
%Y A381078 - For strict multiset partitions with distinct sums (A321469) see A381637.
%Y A381078 - For sets of constant multisets with distinct sums (A381635) see A381716, A381636.
%Y A381078 More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
%Y A381078 A000041 counts integer partitions, strict A000009.
%Y A381078 A000040 lists the primes.
%Y A381078 A003963 gives product of prime indices.
%Y A381078 A055396 gives least prime index, greatest A061395.
%Y A381078 A056239 adds up prime indices, row sums of A112798.
%Y A381078 A122111 represents conjugation in terms of Heinz numbers.
%Y A381078 A265947 counts refinement-ordered pairs of integer partitions.
%Y A381078 Cf. A000720, A001222, A002846, A005117, A025487, A066328, A213242, A213385, A213427, A299201, A299202, A300385.
%K A381078 nonn
%O A381078 1,6
%A A381078 _Gus Wiseman_, Mar 05 2025