A321455 Number of ways to factor n into factors > 1 all having the same sum of prime indices.

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1

Offset

1,4

Comments

Also the number of multiset partitions of the multiset of prime indices of n with equal block-sums.

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. The sum of prime indices of n is A056239(n).

Links

Table of n, a(n) for n=1..87.

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Example

The a(1440) = 6 factorizations into factors all having the same sum of prime indices:
  (10*12*12)
  (5*6*6*8)
  (9*10*16)
  (30*48)
  (36*40)
  (1440)
The a(900) = 5 multiset partitions with equal block-sums:
  {{1,1,2,2,3,3}}
  {{3,3},{1,1,2,2}}
  {{1,2,3},{1,2,3}}
  {{1,3},{1,3},{2,2}}
  {{3},{3},{1,2},{1,2}}

Mathematica

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

Crossrefs

Positions of 1's are A321453. Positions of terms > 1 are A321454.

Cf. A001055, A035470, A056239, A279787, A305551, A321469, A322794, A326515, A326516, A326518, A326534.

Sequence in context: A085424 A088737 A318434 * A096309 A185102 A049419

Adjacent sequences: A321452 A321453 A321454 * A321456 A321457 A321458

Keyword

nonn

Author

Gus Wiseman, Nov 10 2018

Status

approved