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A381453 Number of multisets that can be obtained by choosing a constant integer partition of each prime index of n and taking the multiset union.

%I A381453 #12 Apr 01 2025 11:10:41
%S A381453 1,1,2,1,2,2,3,1,3,2,2,2,4,3,4,1,2,3,4,2,6,2,3,2,3,4,4,3,4,4,2,1,4,2,
%T A381453 6,3,6,4,8,2,2,6,4,2,6,3,4,2,6,3,4,4,5,4,4,3,8,4,2,4,6,2,8,1,8,4,2,2,
%U A381453 6,6,6,3,4,6,6,4,6,8,4,2,5,2,2,6,4,4,8
%N A381453 Number of multisets that can be obtained by choosing a constant integer partition of each prime index of n and taking the multiset union.
%C A381453 First differs from A355733 and A355735 at a(21) = 6, A355733(21) = A355735(21) = 5.
%C A381453 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 A381453 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 A381453 Multisets of constant multisets are generally not transitive. For example, we have arrows: {{1,1},{2}}: {1,1,2} -> {2,2} and {{2,2}}: {2,2} -> {4}, but there is no multiset of constant multisets {1,1,2} -> {4}.
%H A381453 Robert Price, <a href="/A381453/b381453.txt">Table of n, a(n) for n = 1..300</a>
%F A381453 a(A002110(n)) = A381807(n).
%e A381453 The a(21) = 6 multisets are: {2,4}, {1,1,4}, {2,2,2}, {1,1,2,2}, {2,1,1,1,1}, {1,1,1,1,1,1}.
%e A381453 The a(n) partitions for n = 1, 3, 7, 13, 53, 21 (G = 16):
%e A381453   ()  (2)   (4)     (6)       (G)                 (42)
%e A381453       (11)  (22)    (33)      (88)                (411)
%e A381453             (1111)  (222)     (4444)              (222)
%e A381453                     (111111)  (22222222)          (2211)
%e A381453                               (1111111111111111)  (21111)
%e A381453                                                   (111111)
%t A381453 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A381453 Table[Length[Union[Sort/@Join@@@Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@prix[n]]]],{n,nn}]
%Y A381453 Positions of 1 are A000079.
%Y A381453 The strict case is A008966.
%Y A381453 Before sorting we had A355731.
%Y A381453 Choosing divisors instead of constant multisets gives A355733.
%Y A381453 The upper version is A381455, before taking sums A000688.
%Y A381453 Multiset partitions of prime indices:
%Y A381453 - For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
%Y A381453 - For strict multiset partitions (A045778) see A381452.
%Y A381453 - For set multipartitions (A050320) see A381078 (upper), A381454 (lower).
%Y A381453 - For set systems (A050326, zeros A293243) see A381441 (upper).
%Y A381453 - For sets of constant multisets (A050361) see A381715.
%Y A381453 - For strict multiset partitions with distinct sums (A321469) see A381637.
%Y A381453 - For set systems with distinct sums (A381633, zeros A381806) see A381634.
%Y A381453 - For sets of constant multisets with distinct sums (A381635, zeros A381636) see A381716.
%Y A381453 More on multiset partitions into constant blocks: A006171, A279784, A295935.
%Y A381453 A000041 counts integer partitions, strict A000009.
%Y A381453 A000040 lists the primes.
%Y A381453 A003963 gives product of prime indices.
%Y A381453 A055396 gives least prime index, greatest A061395.
%Y A381453 A056239 adds up prime indices, row sums of A112798.
%Y A381453 A122111 represents conjugation in terms of Heinz numbers.
%Y A381453 A265947 counts refinement-ordered pairs of integer partitions.
%Y A381453 Cf. A000720, A000961, A001222, A002577, A018818, A213242, A213385, A213427, A275870, A299200, A300273, A300385.
%K A381453 nonn
%O A381453 1,3
%A A381453 _Gus Wiseman_, Mar 08 2025