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A327781
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Number of integer partitions of n whose LCM is less than n.
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6
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0, 0, 1, 2, 4, 5, 9, 12, 18, 22, 30, 37, 52, 69, 89, 110, 143, 163, 204, 243, 298, 374, 451, 516, 620, 790, 932, 1064, 1243, 1454, 1699, 2365, 2733, 3071, 3524, 3945, 4526, 5600, 6361, 7111, 8057, 9405, 10621, 12836, 14395, 16066, 18047, 19860, 22143, 25748
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OFFSET
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0,4
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LINKS
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EXAMPLE
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The a(2) = 1 through a(8) = 18 partitions:
(11) (21) (22) (41) (33) (61) (44)
(111) (31) (221) (42) (322) (62)
(211) (311) (51) (331) (71)
(1111) (2111) (222) (421) (332)
(11111) (411) (511) (422)
(2211) (2221) (611)
(3111) (3211) (2222)
(21111) (4111) (3221)
(111111) (22111) (3311)
(31111) (4211)
(211111) (5111)
(1111111) (22211)
(32111)
(41111)
(221111)
(311111)
(2111111)
(11111111)
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MAPLE
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a:= proc(m) option remember; local b; b:=
proc(n, i, l) option remember; `if`(n=0, 1,
`if`(i>1, b(n, i-1, l), 0) +(h-> `if`(h<m,
b(n-i, min(n-i, i), h), 0))(ilcm(l, i)))
end: `if`(m>0, b(m$2, 1), 0)
end:
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], LCM@@#<n&]], {n, 30}]
(* Second program: *)
a[m_] := a[m] = Module[{b}, b[n_, i_, l_] := b[n, i, l] =
If[n == 0, 1, If[i>1, b[n, i - 1, l], 0] + Function[h, If[h<m,
b[n - i, Min[n - i, i], h], 0]][LCM[l, i]]]; If[m>0, b[m, m, 1], 0]];
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PROG
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(PARI)
b(m, n)={my(d=divisors(m)); polcoef(1/prod(i=1, #d, 1 - x^d[i] + O(x*x^n)), n)}
a(n)={sum(m=1, n-1, b(m, n)*sum(i=1, (n-1)\m, moebius(i)))} \\ Andrew Howroyd, Oct 09 2019
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CROSSREFS
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The Heinz numbers of these partitions are given by A327776.
Partitions whose LCM is equal to their sum are A074761.
Partitions whose LCM is greater than their sum are A327779.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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