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A328221
Number of integer partitions of n with at least one pair of consecutive divisible parts.
9
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
OFFSET
0,4
COMMENTS
Includes all non-strict partitions.
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], MatchQ[#, {___, x_, y_, ___}/; Divisible[x, y]]&]], {n, 0, 30}]
CROSSREFS
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.
Sequence in context: A370593 A241822 A133732 * A364913 A128215 A325676
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 15 2019
STATUS
approved