A372121 Row sums of A371783 and A371954 (k-quanimous partitions).

1, 3, 4, 9, 8, 22, 16, 42, 41, 74, 57, 183, 102, 233, 263, 463, 298, 875, 491, 1350, 1172, 1775, 1256, 4273, 2225, 4399, 4584, 8049, 4566, 14913, 6843, 18539, 15831, 22894, 18196, 53323, 21638, 48947, 50281, 94500, 44584, 144976, 63262, 173436, 169361, 202153

Offset

1,2

Comments

A finite multiset of numbers is defined to be k-quanimous iff it can be partitioned into k multisets with equal sums. The triangles A371783 and A371954 count k-quanimous partitions.

Links

Table of n, a(n) for n=1..46.

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[Sum[Length[Select[IntegerPartitions[n], Select[facs[Times@@Prime/@#], Length[#]==k&&SameQ@@hwt/@#&]!={}&]], {k, Divisors[n]}], {n, 1, 10}]

Prog

(PARI) T(n, d) = my(v=partitions(n/d), w=List([])); forvec(s=vector(d, i, [1, #v]), listput(w, vecsort(concat(vector(d, i, v[s[i]])))), 1); #Set(w);
a(n) = sumdiv(n, d, T(n, d)); \\ Jinyuan Wang, Feb 13 2025

Crossrefs

Row sums of A371783.

Row sums of A371954.

A000005 counts divisors.

A000041 counts integer partitions.

A002219 (aerated) counts biquanimous partitions, ranks A357976.

A321452 counts quanimous partitions, complement A321451.

A371796 counts quanimous sets, differences A371797.

Cf. A006827, A035470, A064914, A321455, A365543, A371737, A371791, A371795.

Sequence in context: A280616 A317099 A317715 * A305551 A306018 A076120

Adjacent sequences: A372118 A372119 A372120 * A372122 A372123 A372124

Keyword

nonn

Author

Gus Wiseman, Apr 20 2024

Extensions

More terms from Jinyuan Wang, Feb 13 2025

Status

approved