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A220377
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Number of partitions of n into three distinct and mutually relatively prime parts.
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38
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1, 0, 2, 1, 3, 1, 6, 1, 7, 3, 7, 3, 14, 3, 15, 6, 14, 6, 25, 6, 22, 10, 25, 9, 42, 8, 34, 15, 37, 15, 53, 13, 48, 22, 53, 17, 78, 17, 65, 30, 63, 24, 99, 24, 88, 35, 84, 30, 126, 34, 103, 45, 103, 38, 166, 35, 124, 57, 128, 51, 184, 44, 150, 67, 172, 52, 218
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OFFSET
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6,3
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COMMENTS
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The Heinz numbers of these partitions are the intersection of A005117 (strict), A014612 (triples), and A302696 (coprime). - Gus Wiseman, Oct 14 2020
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LINKS
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Fausto A. C. Cariboni, Table of n, a(n) for n = 6..10000 (terms 6..1000 from Seiichi Manyama)
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FORMULA
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a(n > 2) = A307719(n) - 1. - Gus Wiseman, Oct 15 2020
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EXAMPLE
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For n=10 we have three such partitions: 1+2+7, 1+4+5 and 2+3+5.
From Gus Wiseman, Oct 14 2020: (Start)
The a(6) = 1 through a(20) = 15 triples (empty column indicated by dot, A..H = 10..17):
321 . 431 531 532 731 543 751 743 753 754 971 765 B53 875
521 541 651 752 951 853 B51 873 B71 974
721 732 761 B31 871 D31 954 D51 A73
741 851 952 972 A91
831 941 B32 981 B54
921 A31 B41 A71 B72
B21 D21 B43 B81
B52 C71
B61 D43
C51 D52
D32 D61
D41 E51
E31 F41
F21 G31
H21
(End)
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MATHEMATICA
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Table[Length@Select[ IntegerPartitions[ n, {3}], #[[1]] != #[[2]] != #[[3]] && GCD[#[[1]], #[[2]]] == 1 && GCD[#[[1]], #[[3]]] == 1 && GCD[#[[2]], #[[3]]] == 1 &], {n, 6, 100}]
Table[Count[IntegerPartitions[n, {3}], _?(CoprimeQ@@#&&Length[ Union[#]] == 3&)], {n, 6, 100}] (* Harvey P. Dale, May 22 2020 *)
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PROG
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(PARI) a(n)=my(P=partitions(n)); sum(i=1, #P, #P[i]==3&&P[i][1]<P[i][2]&&P[i][2]<P[i][3]&&gcd(P[i][1], P[i][2])==1&&gcd(P[i][1], P[i][3])==1&&gcd(P[i][2], P[i][3])==1) \\ Charles R Greathouse IV, Dec 14 2012
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CROSSREFS
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Cf. A015617, A300815.
A023022 is the 2-part version.
A101271 is the relative prime instead of pairwise coprime version.
A220377*6 is the ordered version.
A305713 counts these partitions of any length, with Heinz numbers A302797.
A307719 is the non-strict version.
A337461 is the non-strict ordered version.
A337563 is the case with no 1's.
A337605 is the pairwise non-coprime instead of pairwise coprime version.
A001399(n-6) counts strict 3-part partitions, with Heinz numbers A007304.
A008284 counts partitions by sum and length, with strict case A008289.
A318717 counts pairwise non-coprime strict partitions.
A326675 ranks pairwise coprime sets.
A327516 counts pairwise coprime partitions.
A337601 counts 3-part partitions whose distinct parts are pairwise coprime.
Cf. A000217, A007360, A023023, A051424, A078374, A087087, A302696, A333227, A337485, A337561.
Sequence in context: A084419 A119606 A034850 * A329696 A145969 A357985
Adjacent sequences: A220374 A220375 A220376 * A220378 A220379 A220380
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KEYWORD
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nonn
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AUTHOR
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Carl Najafi, Dec 13 2012
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STATUS
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approved
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