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A101271 Number of partitions of n into 3 distinct and relatively prime parts.

%I A101271 #24 Jan 24 2021 06:43:42
%S A101271 1,1,2,3,4,5,6,8,9,12,12,16,15,21,20,26,25,33,28,40,36,45,42,56,44,65,
%T A101271 56,70,64,84,66,96,81,100,88,120,90,133,110,132,121,161,120,175,140,
%U A101271 176,156,208,153,220,180,222,196,261,184,280,225,270,240,312,230,341,272
%N A101271 Number of partitions of n into 3 distinct and relatively prime parts.
%C A101271 The Heinz numbers of these partitions are the intersection of A289509 (relatively prime), A005117 (strict), and A014612 (triple). - _Gus Wiseman_, Oct 15 2020
%H A101271 Fausto A. C. Cariboni, <a href="/A101271/b101271.txt">Table of n, a(n) for n = 6..10000</a>
%F A101271 G.f. for the number of partitions of n into m distinct and relatively prime parts is Sum(moebius(k)*x^(m*(m+1)/2*k)/Product(1-x^(i*k), i=1..m), k=1..infinity).
%e A101271 For n=10 we have 4 such partitions: 1+2+7, 1+3+6, 1+4+5 and 2+3+5.
%e A101271 From _Gus Wiseman_, Oct 13 2020: (Start)
%e A101271 The a(6) = 1 through a(18) = 15 triples (A..F = 10..15):
%e A101271   321  421  431  432  532  542  543  643  653  654  754  764  765
%e A101271             521  531  541  632  651  652  743  753  763  854  873
%e A101271                  621  631  641  732  742  752  762  853  863  954
%e A101271                       721  731  741  751  761  843  871  872  972
%e A101271                            821  831  832  851  852  943  953  981
%e A101271                                 921  841  932  861  952  962  A53
%e A101271                                      931  941  942  961  971  A71
%e A101271                                      A21  A31  951  A51  A43  B43
%e A101271                                           B21  A32  B32  A52  B52
%e A101271                                                A41  B41  A61  B61
%e A101271                                                B31  C31  B42  C51
%e A101271                                                C21  D21  B51  D32
%e A101271                                                          C32  D41
%e A101271                                                          C41  E31
%e A101271                                                          D31  F21
%e A101271                                                          E21
%e A101271 (End)
%p A101271 m:=3: with(numtheory): g:=sum(mobius(k)*x^(m*(m+1)/2*k)/Product(1-x^(i*k),i=1..m),k=1..20): gser:=series(g,x=0,80): seq(coeff(gser,x^n),n=6..77); # _Emeric Deutsch_, May 31 2005
%t A101271 Table[Length[Select[IntegerPartitions[n,{3}],UnsameQ@@#&&GCD@@#==1&]],{n,6,50}] (* _Gus Wiseman_, Oct 13 2020 *)
%Y A101271 Cf. A023024-A023030, A000742-A000743, A023031-A023035.
%Y A101271 A000741 is the ordered non-strict version.
%Y A101271 A001399(n-6) does not require relative primality.
%Y A101271 A023022 counts pairs instead of triples.
%Y A101271 A023023 is the not necessarily strict version.
%Y A101271 A078374 counts these partitions of any length, with Heinz numbers A302796.
%Y A101271 A101271*6 is the ordered version.
%Y A101271 A220377 is the pairwise coprime instead of relatively prime version.
%Y A101271 A284825 counts the case that is pairwise non-coprime also.
%Y A101271 A337605 is the pairwise non-coprime instead of relatively prime version.
%Y A101271 A008289 counts strict partitions by sum and length.
%Y A101271 A007304 gives the Heinz numbers of 3-part strict partitions.
%Y A101271 A307719 counts 3-part pairwise coprime partitions.
%Y A101271 A337601 counts 3-part partitions whose distinct parts are pairwise coprime.
%Y A101271 Cf. A000010, A000217, A000837, A007360, A014612, A055684, A289509, A332004, A337452, A337563.
%K A101271 easy,nonn
%O A101271 6,3
%A A101271 _Vladeta Jovovic_, Dec 19 2004
%E A101271 More terms from _Emeric Deutsch_, May 31 2005