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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 July 8 04:16 EDT 2020. Contains 335504 sequences. (Running on oeis4.)