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