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A371734
Maximal length of a factorization of n into factors > 1 all having different sums of prime indices.
2
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
OFFSET
1,6
COMMENTS
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.
Factorizations into factors > 1 all having different sums of prime indices are counted by A321469.
EXAMPLE
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.
MATHEMATICA
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}]
CROSSREFS
For set partitions of binary indices 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 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.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
Sequence in context: A345935 A214715 A244145 * A375272 A086435 A266226
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 13 2024
STATUS
approved