OFFSET
0,4
EXAMPLE
The a(1) = 1 through a(9) = 20 partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(21) (31) (32) (42) (43) (53) (54)
(41) (51) (52) (62) (63)
(221) (411) (61) (71) (72)
(311) (322) (332) (81)
(2111) (331) (521) (432)
(421) (611) (441)
(511) (5111) (522)
(2221) (531)
(3211) (621)
(4111) (711)
(22111) (3222)
(31111) (4221)
(211111) (4311)
(5211)
(6111)
(22221)
(42111)
(51111)
(411111)
A complete list of all multiset partitions of the partition (2111) into two or more blocks is: ((1)(112)), ((2)(111)), ((11)(12)), ((1)(1)(12)), ((1)(2)(11)), ((1)(1)(1)(2)). None of these has equal block-sums, so (2111) is counted toward a(5).
On the other hand, the partition (321) can be partitioned as ((12)(3)), which has two or more blocks and equal block-sums, so (321) is not counted toward a(6).
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[IntegerPartitions[n], Length[Select[facs[Times@@Prime/@#], SameQ@@hwt/@#&]]==1&]], {n, 10}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 10 2018
EXTENSIONS
a(33)-a(50) from Alois P. Heinz, Nov 11 2018
STATUS
approved