A381455 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into a multiset of constant multisets.

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1, 1

Offset: 1

Author

Gus Wiseman, Mar 06 2025

Keywords

• nonn

Comments

First differs from A000688 at a(144) = 9, A000688(144) = 10.

First differs from A295879 at a(128) = 15, A295879(128) = 13.

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 prime powers > 1.

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}.

Examples

			The prime indices of 36 are {1,1,2,2}, with the following 4 partitions into a multiset of constant multisets:
  {{1,1},{2,2}}
  {{1},{1},{2,2}}
  {{2},{2},{1,1}}
  {{1},{1},{2},{2}}
with block-sums: {2,4}, {1,1,4}, {2,2,2}, {1,1,2,2}, which are all different, so a(36) = 4.
The prime indices of 144 are {1,1,1,1,2,2}, with the following 10 partitions into a multiset of constant multisets:
  {{2,2},{1,1,1,1}}
  {{1},{2,2},{1,1,1}}
  {{2},{2},{1,1,1,1}}
  {{1,1},{1,1},{2,2}}
  {{1},{1},{1,1},{2,2}}
  {{1},{2},{2},{1,1,1}}
  {{2},{2},{1,1},{1,1}}
  {{1},{1},{1},{1},{2,2}}
  {{1},{1},{2},{2},{1,1}}
  {{1},{1},{1},{1},{2},{2}}
with block-sums: {4,4}, {1,3,4}, {2,2,4}, {2,2,4}, {1,1,2,4}, {1,2,2,3}, {2,2,2,2}, {1,1,1,1,4}, {1,1,2,2,2}, {1,1,1,1,2,2}, of which 9 are distinct, so a(144) = 9.
The a(n) partitions for n = 4, 8, 16, 32, 36, 64, 72, 128:
  (2)   (3)    (4)     (5)      (42)    (6)       (43)     (7)
  (11)  (21)   (22)    (32)     (222)   (33)      (322)    (43)
        (111)  (31)    (41)     (411)   (42)      (421)    (52)
               (211)   (221)    (2211)  (51)      (2221)   (61)
               (1111)  (311)            (222)     (4111)   (322)
                       (2111)           (321)     (22111)  (331)
                       (11111)          (411)              (421)
                                        (2211)             (511)
                                        (3111)             (2221)
                                        (21111)            (3211)
                                        (111111)           (4111)
                                                           (22111)
                                                           (31111)
                                                           (211111)
                                                           (1111111)
		

Links

• Robert Price, Table of n, a(n) for n = 1..1000

Crossrefs

Before taking sums we had A000688.

Positions of 1 are A005117.

There is a chain from the prime indices of n to a singleton iff n belongs to A300273.

The lower version is A381453.

For distinct blocks we have A381715, before sum A050361.

For distinct block-sums we have A381716, before sums A381635 (zeros A381636).

Other 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) see A381441 (upper).

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

- For set systems with distinct sums (A381633) see A381634, A293243.

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, A001222, A002577, A002846, A018818, A213242, A213427, A275870, A289078, A299200, A299202, A300385.

Programs

Mathematica

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[sqfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Table[Length[Union[Sort[hwt/@#]&/@sqfacs[n]]],{n,100}]

Formula

a(s) = 1 for any squarefree number s.

a(p^k) = A000041(k) for any prime p.