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A078374 Number of partitions of n into distinct and relatively prime parts.

%I A078374 #20 Oct 22 2020 06:49:41
%S A078374 1,0,1,1,2,2,4,4,6,7,11,10,17,17,23,26,37,36,53,53,70,77,103,103,139,
%T A078374 147,184,199,255,260,339,358,435,474,578,611,759,810,963,1045,1259,
%U A078374 1331,1609,1726,2015,2200,2589,2762,3259,3509,4058,4416,5119,5488,6364,6882
%N A078374 Number of partitions of n into distinct and relatively prime parts.
%C A078374 The Heinz numbers of these partitions are given by A302796, which is the intersection of A005117 (strict) and A289509 (relatively prime). - _Gus Wiseman_, Oct 18 2020
%H A078374 Seiichi Manyama, <a href="/A078374/b078374.txt">Table of n, a(n) for n = 1..10000</a>
%H A078374 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A078374 Moebius transform of A000009.
%F A078374 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = Product_{n>=1} (1 + x^n). - _Ilya Gutkovskiy_, Apr 26 2017
%e A078374 From _Gus Wiseman_, Oct 18 2020: (Start)
%e A078374 The a(1) = 1 through a(13) = 17 partitions (empty column indicated by dot, A = 10, B = 11, C = 12):
%e A078374   1   .  21   31   32   51    43    53    54    73     65     75     76
%e A078374                    41   321   52    71    72    91     74     B1     85
%e A078374                               61    431   81    532    83     543    94
%e A078374                               421   521   432   541    92     651    A3
%e A078374                                           531   631    A1     732    B2
%e A078374                                           621   721    542    741    C1
%e A078374                                                 4321   632    831    643
%e A078374                                                        641    921    652
%e A078374                                                        731    5421   742
%e A078374                                                        821    6321   751
%e A078374                                                        5321          832
%e A078374                                                                      841
%e A078374                                                                      931
%e A078374                                                                      A21
%e A078374                                                                      5431
%e A078374                                                                      6421
%e A078374                                                                      7321
%e A078374 (End)
%t A078374 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&GCD@@#==1&]],{n,15}] (* _Gus Wiseman_, Oct 18 2020 *)
%Y A078374 Cf. A047966.
%Y A078374 A000837 is the not necessarily strict version.
%Y A078374 A302796 gives the Heinz numbers of these partitions.
%Y A078374 A305713 is the pairwise coprime instead of relatively prime version.
%Y A078374 A332004 is the ordered version.
%Y A078374 A337452 is the case without 1's.
%Y A078374 A000009 counts strict partitions.
%Y A078374 A000740 counts relatively prime compositions.
%Y A078374 Cf. A007359, A101268, A289508, A289509, A291166, A298748, A337451, A337485, A337451, A337561, A337563.
%K A078374 nonn
%O A078374 1,5
%A A078374 _Vladeta Jovovic_, Dec 24 2002