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.

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, 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, 6, 6, 6, 3, 4, 6, 6, 4, 6, 8, 4, 2, 5, 2, 2, 6, 4, 4, 8

Offset

1,3

Comments

First differs from A355733 and A355735 at a(21) = 6, A355733(21) = A355735(21) = 5.

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.

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).

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}.

Links

Robert Price, Table of n, a(n) for n = 1..300

Formula

a(A002110(n)) = A381807(n).

Example

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}.
The a(n) partitions for n = 1, 3, 7, 13, 53, 21 (G = 16):
  ()  (2)   (4)     (6)       (G)                 (42)
      (11)  (22)    (33)      (88)                (411)
            (1111)  (222)     (4444)              (222)
                    (111111)  (22222222)          (2211)
                              (1111111111111111)  (21111)
                                                  (111111)

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[Union[Sort/@Join@@@Tuples[Select[IntegerPartitions[#], SameQ@@#&]&/@prix[n]]]], {n, nn}]

Crossrefs

Positions of 1 are A000079.

The strict case is A008966.

Before sorting we had A355731.

Choosing divisors instead of constant multisets gives A355733.

The upper version is A381455, before taking sums A000688.

Multiset partitions of prime indices:

- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).

- For strict multiset partitions (A045778) see A381452.

- For set multipartitions (A050320) see A381078 (upper), A381454 (lower).

- For set systems (A050326, zeros A293243) see A381441 (upper).

- For sets of constant multisets (A050361) see A381715.

- For strict multiset partitions with distinct sums (A321469) see A381637.

- For set systems with distinct sums (A381633, zeros A381806) see A381634.

- For sets of constant multisets with distinct sums (A381635, zeros A381636) see A381716.

More on multiset partitions into constant blocks: A006171, A279784, A295935.

A000041 counts integer partitions, strict A000009.

A000040 lists the primes.

A003963 gives product of prime indices.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A122111 represents conjugation in terms of Heinz numbers.

A265947 counts refinement-ordered pairs of integer partitions.

Cf. A000720, A000961, A001222, A002577, A018818, A213242, A213385, A213427, A275870, A299200, A300273, A300385.

Sequence in context: A000374 A355735 A355733 * A256757 A333860 A277314

Adjacent sequences: A381450 A381451 A381452 * A381454 A381455 A381456

Keyword

nonn

Author

Gus Wiseman, Mar 08 2025

Status

approved