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%I #34 Jan 09 2024 08:45:42
%S 1,1,2,2,4,4,6,6,10,8,14,12,16,16,24,18,30,24,32,30,44,32,50,42,54,48,
%T 70,48,80,64,80,72,96,72,114,90,112,96,140,96,154,120,144,132,184,128,
%U 196,150,192,168,234,162,240,192,240,210,290,192,310,240,288,256,336,240,374
%N Number of partitions of n into 3 unordered relatively prime parts.
%H Fausto A. C. Cariboni, <a href="/A023023/b023023.txt">Table of n, a(n) for n = 3..10000</a>
%H Mohamed El Bachraoui, <a href="https://www.fq.math.ca/Papers1/46_47-4/Bachraoui.pdf">Relatively Prime Partitions with Two and Three Parts</a>, Fibonacci Quart. 46/47 (2008/2009), no. 4, 341-345.
%F G.f. for the number of partitions of n into m unordered relatively prime parts is Sum(moebius(k)*x^(m*k)/Product(1-x^(i*k), i=1..m), k=1..infinity). - _Vladeta Jovovic_, Dec 21 2004
%F a(n) = (n^2/12)*Product_{prime p|n} (1 - 1/p^2) = A007434(n)/12 for n > 3 (proved by Mohamed El Bachraoui). [_Jonathan Sondow_, May 27 2009]
%F a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} floor(1/gcd(i,k,n-i-k)). - _Wesley Ivan Hurt_, Jan 02 2021
%e From _Gus Wiseman_, Oct 08 2020: (Start)
%e The a(3) = 1 through a(13) = 14 triples (A = 10, B = 11):
%e 111 211 221 321 322 332 432 433 443 543 544
%e 311 411 331 431 441 532 533 552 553
%e 421 521 522 541 542 651 643
%e 511 611 531 631 551 732 652
%e 621 721 632 741 661
%e 711 811 641 831 733
%e 722 921 742
%e 731 A11 751
%e 821 832
%e 911 841
%e 922
%e 931
%e A21
%e B11
%e (End)
%t Table[Length[Select[IntegerPartitions[n,{3}],GCD@@#==1&]],{n,3,50}] (* _Gus Wiseman_, Oct 08 2020 *)
%Y Cf. A023024-A023030, A000742-A000743, A023032-A023035.
%Y A000741 is the ordered version.
%Y A000837 counts these partitions of any length.
%Y A001399(n-3) does not require relative primality.
%Y A023022 is the 2-part version.
%Y A101271 is the strict case.
%Y A284825 counts the case that is also pairwise non-coprime.
%Y A289509 intersected with A014612 gives the Heinz numbers.
%Y A307719 is the pairwise coprime instead of relatively prime version.
%Y A337599 is the pairwise non-coprime instead of relative prime version.
%Y A008284 counts partitions by sum and length.
%Y A078374 counts relatively prime strict partitions.
%Y A337601 counts 3-part partitions whose distinct parts are pairwise coprime.
%Y Cf. A000010, A000217, A007434, A055684, A078374, A200976, A220377, A302698, A327516, A337563, A337600, A337605.
%K nonn
%O 3,3
%A _David W. Wilson_
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