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 A328172 Number of integer partitions of n with all pairs of consecutive parts relatively prime. 17
 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 (list; graph; refs; listen; history; text; internal format)
 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 Alois P. Heinz, Table of n, a(n) for n = 0..1000 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}] CROSSREFS The case of compositions is A167606. The strict case is A328188. The Heinz numbers of these partitions are given by A328335. Cf. A000837, A018783, A178470, A328028, A328170, A328171, A328187, A328188 A328220. Sequence in context: A308632 A137606 A320224 * A239468 A119793 A181436 Adjacent sequences:  A328169 A328170 A328171 * A328173 A328174 A328175 KEYWORD nonn AUTHOR Gus Wiseman, Oct 12 2019 STATUS approved

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Last modified February 22 14:51 EST 2020. Contains 332137 sequences. (Running on oeis4.)