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A328172
Number of integer partitions of n with all pairs of consecutive parts relatively prime.
23
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
OFFSET
0,3
COMMENTS
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.
LINKS
EXAMPLE
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)
MAPLE
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, {}):
seq(a(n), n=0..60); # Alois P. Heinz, Oct 13 2019
MATHEMATICA
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, {}];
a /@ Range[0, 60] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)
CROSSREFS
The case of compositions is A167606.
The strict case is A328188.
The Heinz numbers of these partitions are given by A328335.
Sequence in context: A137606 A320224 A347461 * A239468 A336815 A355979
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 12 2019
STATUS
approved