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A101271
Number of partitions of n into 3 distinct and relatively prime parts.
22
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, 56, 70, 64, 84, 66, 96, 81, 100, 88, 120, 90, 133, 110, 132, 121, 161, 120, 175, 140, 176, 156, 208, 153, 220, 180, 222, 196, 261, 184, 280, 225, 270, 240, 312, 230, 341, 272
OFFSET
6,3
COMMENTS
The Heinz numbers of these partitions are the intersection of A289509 (relatively prime), A005117 (strict), and A014612 (triple). - Gus Wiseman, Oct 15 2020
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 6..10000
FORMULA
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).
EXAMPLE
For n=10 we have 4 such partitions: 1+2+7, 1+3+6, 1+4+5 and 2+3+5.
From Gus Wiseman, Oct 13 2020: (Start)
The a(6) = 1 through a(18) = 15 triples (A..F = 10..15):
321 421 431 432 532 542 543 643 653 654 754 764 765
521 531 541 632 651 652 743 753 763 854 873
621 631 641 732 742 752 762 853 863 954
721 731 741 751 761 843 871 872 972
821 831 832 851 852 943 953 981
921 841 932 861 952 962 A53
931 941 942 961 971 A71
A21 A31 951 A51 A43 B43
B21 A32 B32 A52 B52
A41 B41 A61 B61
B31 C31 B42 C51
C21 D21 B51 D32
C32 D41
C41 E31
D31 F21
E21
(End)
MAPLE
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
MATHEMATICA
Table[Length[Select[IntegerPartitions[n, {3}], UnsameQ@@#&&GCD@@#==1&]], {n, 6, 50}] (* Gus Wiseman, Oct 13 2020 *)
CROSSREFS
A000741 is the ordered non-strict version.
A001399(n-6) does not require relative primality.
A023022 counts pairs instead of triples.
A023023 is the not necessarily strict version.
A078374 counts these partitions of any length, with Heinz numbers A302796.
A101271*6 is the ordered version.
A220377 is the pairwise coprime instead of relatively prime version.
A284825 counts the case that is pairwise non-coprime also.
A337605 is the pairwise non-coprime instead of relatively prime version.
A008289 counts strict partitions by sum and length.
A007304 gives the Heinz numbers of 3-part strict partitions.
A307719 counts 3-part pairwise coprime partitions.
A337601 counts 3-part partitions whose distinct parts are pairwise coprime.
Sequence in context: A100054 A330193 A372667 * A093110 A165707 A052063
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Dec 19 2004
EXTENSIONS
More terms from Emeric Deutsch, May 31 2005
STATUS
approved