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 A078374 Number of partitions of n into distinct and relatively prime parts. 68

%I

%S 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 147,184,199,255,260,339,358,435,474,578,611,759,810,963,1045,1259,

%U 1331,1609,1726,2015,2200,2589,2762,3259,3509,4058,4416,5119,5488,6364,6882

%N Number of partitions of n into distinct and relatively prime parts.

%C 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 Seiichi Manyama, <a href="/A078374/b078374.txt">Table of n, a(n) for n = 1..10000</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F Moebius transform of A000009.

%F 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 From _Gus Wiseman_, Oct 18 2020: (Start)

%e The a(1) = 1 through a(13) = 17 partitions (empty column indicated by dot, A = 10, B = 11, C = 12):

%e 1 . 21 31 32 51 43 53 54 73 65 75 76

%e 41 321 52 71 72 91 74 B1 85

%e 61 431 81 532 83 543 94

%e 421 521 432 541 92 651 A3

%e 531 631 A1 732 B2

%e 621 721 542 741 C1

%e 4321 632 831 643

%e 641 921 652

%e 731 5421 742

%e 821 6321 751

%e 5321 832

%e 841

%e 931

%e A21

%e 5431

%e 6421

%e 7321

%e (End)

%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&GCD@@#==1&]],{n,15}] (* _Gus Wiseman_, Oct 18 2020 *)

%Y Cf. A047966.

%Y A000837 is the not necessarily strict version.

%Y A302796 gives the Heinz numbers of these partitions.

%Y A305713 is the pairwise coprime instead of relatively prime version.

%Y A332004 is the ordered version.

%Y A337452 is the case without 1's.

%Y A000009 counts strict partitions.

%Y A000740 counts relatively prime compositions.

%Y Cf. A007359, A101268, A289508, A289509, A291166, A298748, A337451, A337485, A337451, A337561, A337563.

%K nonn

%O 1,5

%A _Vladeta Jovovic_, Dec 24 2002

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Last modified April 10 19:06 EDT 2021. Contains 342853 sequences. (Running on oeis4.)