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A321455
Number of ways to factor n into factors > 1 all having the same sum of prime indices.
31
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).
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.
Sequence in context: A085424 A088737 A318434 * A096309 A185102 A049419
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 10 2018
STATUS
approved