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A328221
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Number of integer partitions of n with at least one pair of consecutive divisible parts.
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9
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0, 0, 1, 2, 4, 5, 10, 12, 20, 26, 38, 51, 73, 92, 126, 166, 219, 283, 369, 470, 604, 763, 968, 1217, 1534, 1907, 2376, 2944, 3640, 4476, 5501, 6723, 8212, 9986, 12130, 14682, 17748, 21376, 25717, 30847, 36959, 44152, 52688, 62714, 74557, 88440, 104775, 123878
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OFFSET
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0,4
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COMMENTS
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Includes all non-strict partitions.
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LINKS
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EXAMPLE
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The a(2) = 1 through a(8) = 20 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) (321) (511) (422)
(411) (2221) (431)
(2211) (3211) (521)
(3111) (4111) (611)
(21111) (22111) (2222)
(111111) (31111) (3221)
(211111) (3311)
(1111111) (4211)
(5111)
(22211)
(32111)
(41111)
(221111)
(311111)
(2111111)
(11111111)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], MatchQ[#, {___, x_, y_, ___}/; Divisible[x, y]]&]], {n, 0, 30}]
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CROSSREFS
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The complement is counted by A328171.
Partitions whose consecutive parts are relatively prime are A328172.
Partitions with no pair of consecutive parts relatively prime are A328187.
Numbers without consecutive divisible proper divisors are A328028.
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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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