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 A023023 Number of partitions of n into 3 unordered relatively prime parts. 22
 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, 70, 48, 80, 64, 80, 72, 96, 72, 114, 90, 112, 96, 140, 96, 154, 120, 144, 132, 184, 128, 196, 150, 192, 168, 234, 162, 240, 192, 240, 210, 290, 192, 310, 240, 288, 256, 336, 240, 374 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,3 LINKS Fausto A. C. Cariboni, Table of n, a(n) for n = 3..10000 Mohamed El Bachraoui, Relatively Prime Partitions with Two and Three Parts, Fibonacci Quart. 46/47 (2008/2009), no. 4, 341-345. FORMULA 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 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] 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 EXAMPLE From Gus Wiseman, Oct 08 2020: (Start) The a(3) = 1 through a(13) = 14 triples (A = 10, B = 11): 111 211 221 321 322 332 432 433 443 543 544 311 411 331 431 441 532 533 552 553 421 521 522 541 542 651 643 511 611 531 631 551 732 652 621 721 632 741 661 711 811 641 831 733 722 921 742 731 A11 751 821 832 911 841 922 931 A21 B11 (End) MATHEMATICA Table[Length[Select[IntegerPartitions[n, {3}], GCD@@#==1&]], {n, 3, 50}] (* Gus Wiseman, Oct 08 2020 *) CROSSREFS Cf. A023024-A023030, A000742-A000743, A023032-A023035. A000741 is the ordered version. A000837 counts these partitions of any length. A001399(n-3) does not require relative primality. A023022 is the 2-part version. A101271 is the strict case. A284825 counts the case that is also pairwise non-coprime. A289509 intersected with A014612 gives the Heinz numbers. A307719 is the pairwise coprime instead of relatively prime version. A337599 is the pairwise non-coprime instead of relative prime version. A008284 counts partitions by sum and length. A078374 counts relatively prime strict partitions. A337601 counts 3-part partitions whose distinct parts are pairwise coprime. Cf. A000010, A000217, A007434, A055684, A078374, A200976, A220377, A302698, A327516, A337563, A337600, A337605. Sequence in context: A340280 A340279 A343100 * A184157 A008643 A008644 Adjacent sequences: A023020 A023021 A023022 * A023024 A023025 A023026 KEYWORD nonn AUTHOR David W. Wilson STATUS approved

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Last modified September 16 02:52 EDT 2024. Contains 375959 sequences. (Running on oeis4.)