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A321451
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Number of integer partitions of n that cannot be partitioned into two or more blocks with equal sums.
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15
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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
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OFFSET
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0,4
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LINKS
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FORMULA
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EXAMPLE
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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).
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MATHEMATICA
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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}]
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CROSSREFS
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Cf. A000041, A265947, A276024, A279787, A305551, A306017, A317141, A320322, A321452, A321453, A321454, A321455.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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