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 A220377 Number of partitions of n into three distinct and mutually relatively prime parts. 36
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 6,3 COMMENTS The Heinz numbers of these partitions are the intersection of A005117 (strict), A014612 (triples), and A302696 (coprime). - Gus Wiseman, Oct 14 2020 LINKS Fausto A. C. Cariboni, Table of n, a(n) for n = 6..10000 (terms 6..1000 from Seiichi Manyama) FORMULA a(n > 2) = A307719(n) - 1. - Gus Wiseman, Oct 15 2020 EXAMPLE 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) MATHEMATICA 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 *) PROG (PARI) a(n)=my(P=partitions(n)); sum(i=1, #P, #P[i]==3&&P[i][1]

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Last modified April 20 06:59 EDT 2021. Contains 343125 sequences. (Running on oeis4.)