A321451 Number of integer partitions of n that cannot be partitioned into two or more blocks with equal sums.

1, 1, 1, 2, 2, 6, 4, 14, 8, 20, 16, 55, 22, 100, 45, 108, 64, 296, 93, 489, 145, 447, 241, 1254, 284, 1692, 487, 1492, 627, 4564, 811, 6841, 1172, 4531, 1744, 12260, 1970, 21636, 3103, 12193, 3719, 44582, 4645, 63260, 6417, 29947, 8987, 124753, 9784, 162107, 14247

Offset: 0

Author

Gus Wiseman, Nov 10 2018

Keywords

• nonn

Examples

			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).
		

Crossrefs

Cf. A000041, A265947, A276024, A279787, A305551, A306017, A317141, A320322, A321452, A321453, A321454, A321455.

Programs

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}]

Formula

a(n) = A000041(n) - A321452(n).

Extensions

a(33)-a(50) from Alois P. Heinz, Nov 11 2018