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A023023 Number of partitions of n into 3 unordered relatively prime parts.

%I A023023 #35 Jan 05 2025 19:51:34
%S A023023 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 A023023 70,48,80,64,80,72,96,72,114,90,112,96,140,96,154,120,144,132,184,128,
%U A023023 196,150,192,168,234,162,240,192,240,210,290,192,310,240,288,256,336,240,374
%N A023023 Number of partitions of n into 3 unordered relatively prime parts.
%H A023023 Fausto A. C. Cariboni, <a href="/A023023/b023023.txt">Table of n, a(n) for n = 3..10000</a>
%H A023023 Mohamed El Bachraoui, <a href="https://web.archive.org/web/2024*/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 A023023 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 A023023 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 A023023 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 A023023 From _Gus Wiseman_, Oct 08 2020: (Start)
%e A023023 The a(3) = 1 through a(13) = 14 triples (A = 10, B = 11):
%e A023023   111   211   221   321   322   332   432   433   443   543   544
%e A023023               311   411   331   431   441   532   533   552   553
%e A023023                           421   521   522   541   542   651   643
%e A023023                           511   611   531   631   551   732   652
%e A023023                                       621   721   632   741   661
%e A023023                                       711   811   641   831   733
%e A023023                                                   722   921   742
%e A023023                                                   731   A11   751
%e A023023                                                   821         832
%e A023023                                                   911         841
%e A023023                                                               922
%e A023023                                                               931
%e A023023                                                               A21
%e A023023                                                               B11
%e A023023 (End)
%t A023023 Table[Length[Select[IntegerPartitions[n,{3}],GCD@@#==1&]],{n,3,50}] (* _Gus Wiseman_, Oct 08 2020 *)
%Y A023023 Cf. A023024-A023030, A000742-A000743, A023032-A023035.
%Y A023023 A000741 is the ordered version.
%Y A023023 A000837 counts these partitions of any length.
%Y A023023 A001399(n-3) does not require relative primality.
%Y A023023 A023022 is the 2-part version.
%Y A023023 A101271 is the strict case.
%Y A023023 A284825 counts the case that is also pairwise non-coprime.
%Y A023023 A289509 intersected with A014612 gives the Heinz numbers.
%Y A023023 A307719 is the pairwise coprime instead of relatively prime version.
%Y A023023 A337599 is the pairwise non-coprime instead of relative prime version.
%Y A023023 A008284 counts partitions by sum and length.
%Y A023023 A078374 counts relatively prime strict partitions.
%Y A023023 A337601 counts 3-part partitions whose distinct parts are pairwise coprime.
%Y A023023 Cf. A000010, A000217, A007434, A055684, A078374, A200976, A220377, A302698, A327516, A337563, A337600, A337605.
%K A023023 nonn
%O A023023 3,3
%A A023023 _David W. Wilson_