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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. Sum of prime indices is given by A056239.

For same sums instead of different sums we have A371733.

A321142 and A371794 count non-biquanimous strict partitions.

A371789 counts non-quanimous sets, differences A371790.

A371796 counts quanimous sets, differences A371797.

Cf. A035470, A279787, A305551, A321142, A322794, A326515, A326518, A326534, A336137, A371783, A371789, A371791.

For set partitions instead of factorizations binary indices we have A000120, same sums A371735.

allocated for Gus WisemanMaximal length of a factorization of n into factors > 1 all having different sums of prime indices.

0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 3, 2, 2, 1, 3, 2, 2, 2

1,6

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.

Factorizations into factors > 1 all having different sums of prime indices are counted by A321469.

Gus Wiseman, <a href="/A038041/a038041.txt">Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.</a>

The factorizations of 90 of this type are (2*3*15), (2*5*9), (2*45), (3*30), (5*18), (6*15), (90), so a(90) = 3.

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];

hwt[n_]:=Total[Cases[FactorInteger[n], {p_, k_}:>PrimePi[p]*k]];

Table[Max[Length/@Select[facs[n], UnsameQ@@hwt/@#&]], {n, 100}]

For set partitions instead of factorizations we have A000120, same sums A371735.

Positions of 1's are A000430.

Positions of terms > 1 are A080257.

Factorizations of this type are counted by A321469, same sums A321455.

For same sums instead of different sums we have A371733.

A001055 counts factorizations.

A002219 (aerated) counts biquanimous partitions, ranks A357976.

A112798 lists prime indices, reverse A296150, length A001222, sum A056239.

A321142 and A371794 count non-biquanimous strict partitions.

A321451 counts non-quanimous partitions, ranks A321453.

A321452 counts quanimous partitions, ranks A321454.

A371789 counts non-quanimous sets, differences A371790.

A371796 counts quanimous sets, differences A371797.

Cf. A035470, A279787, A305551, A322794, A326515, A326518, A326534, A336137, A371783, A371791.

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Gus Wiseman, Apr 13 2024

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