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A328172
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Number of integer partitions of n with all pairs of consecutive parts relatively prime.
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23
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1, 1, 2, 3, 4, 6, 7, 10, 12, 16, 19, 24, 28, 36, 43, 51, 62, 74, 87, 104, 122, 143, 169, 195, 227, 260, 302, 346, 397, 455, 521, 599, 686, 780, 889, 1001, 1138, 1286, 1454, 1638, 1846, 2076, 2330, 2614, 2929, 3280, 3666, 4093, 4565, 5085, 5667, 6300, 7002
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OFFSET
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0,3
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COMMENTS
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Except for any number of 1's, these partitions must be strict. The fully strict case is A328188.
Partitions with no consecutive pair of parts relatively prime are A328187, with strict case A328220.
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LINKS
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EXAMPLE
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The a(1) = 1 through a(8) = 12 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (31) (32) (51) (43) (53)
(111) (211) (41) (321) (52) (71)
(1111) (311) (411) (61) (431)
(2111) (3111) (511) (521)
(11111) (21111) (3211) (611)
(111111) (4111) (5111)
(31111) (32111)
(211111) (41111)
(1111111) (311111)
(2111111)
(11111111)
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MAPLE
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b:= proc(n, i, s) option remember; `if`(n=0 or i=1, 1,
`if`(andmap(j-> igcd(i, j)=1, s), b(n-i, min(n-i, i-1),
numtheory[factorset](i)), 0)+b(n, i-1, s))
end:
a:= n-> b(n$2, {}):
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], !MatchQ[#, {___, x_, y_, ___}/; GCD[x, y]>1]&]], {n, 0, 30}]
(* Second program: *)
b[n_, i_, s_] := b[n, i, s] = If[n == 0 || i == 1, 1,
If[AllTrue[s, GCD[i, #] == 1&], b[n - i, Min[n - i, i - 1],
FactorInteger[i][[All, 1]]], 0] + b[n, i - 1, s]];
a[n_] := b[n, n, {}];
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CROSSREFS
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The case of compositions is A167606.
The Heinz numbers of these partitions are given by A328335.
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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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