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Number of partitions of n into 3 distinct and relatively prime parts.
22

%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