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A371783
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Irregular triangle read by rows where T(n,d) is the number of integer partitions of n that can be partitioned into d blocks with equal sums, with d ranging over all divisors d|n.
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31
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1, 2, 1, 3, 1, 5, 3, 1, 7, 1, 11, 6, 4, 1, 15, 1, 22, 14, 5, 1, 30, 10, 1, 42, 25, 6, 1, 56, 1, 77, 53, 30, 15, 7, 1, 101, 1, 135, 89, 8, 1, 176, 65, 21, 1
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OFFSET
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1,2
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COMMENTS
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LINKS
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EXAMPLE
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Triangle begins:
1
2 1
3 1
5 3 1
7 1
11 6 4 1
15 1
22 14 5 1
30 10 1
42 25 6 1
56 1
77 53 30 15 7 1
101 1
135 89 8 1
176 65 21 1
Row n = 6 counts the following partitions:
(6) (33) (222) (111111)
(33) (321) (2211)
(42) (2211) (21111)
(51) (3111) (111111)
(222) (21111)
(321) (111111)
(411)
(2211)
(3111)
(21111)
(111111)
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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], Select[facs[Times@@Prime/@#], Length[#]==k&&SameQ@@hwt/@#&]!={}&]], {n, 1, 8}, {k, Divisors[n]}]
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CROSSREFS
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A371781 lists numbers with biquanimous prime signature, complement A371782.
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KEYWORD
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nonn,tabf,more
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AUTHOR
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STATUS
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approved
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