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 A338317 Number of integer partitions of n with no 1's and pairwise coprime distinct parts, where a singleton is always considered coprime. 3
 1, 0, 1, 1, 2, 2, 3, 4, 5, 6, 7, 11, 11, 16, 16, 19, 25, 32, 34, 44, 46, 53, 66, 80, 88, 101, 116, 132, 150, 180, 204, 229, 254, 287, 331, 366, 426, 473, 525, 584, 662, 742, 835, 922, 1013, 1128, 1262, 1408, 1555, 1711, 1894, 2080, 2297, 2555, 2806, 3064, 3376 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS Table of n, a(n) for n=0..56. FORMULA The Heinz numbers of these partitions are given by A338316. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions. EXAMPLE The a(2) = 1 through a(12) = 11 partitions (A = 10, B = 11, C = 12): 2 3 4 5 6 7 8 9 A B C 22 32 33 43 44 54 55 65 66 222 52 53 72 73 74 75 322 332 333 433 83 444 2222 522 532 92 543 3222 3322 443 552 22222 533 732 722 3333 3332 5322 5222 33222 32222 222222 MATHEMATICA Table[Length[Select[IntegerPartitions[n], !MemberQ[#, 1]&&(SameQ@@#||CoprimeQ@@Union[#])&]], {n, 0, 15}] CROSSREFS A007359 (A302568) gives the strict case. A101268 (A335235) gives pairwise coprime or singleton compositions. A200976 (A338318) gives the pairwise non-coprime instead of coprime version. A304709 (A304711) gives partitions whose distinct parts are pairwise coprime, with strict case A305713 (A302797). A304712 (A338331) allows 1's, with strict version A007360 (A302798). A327516 (A302696) gives pairwise coprime partitions. A328673 (A328867) gives partitions with no distinct relatively prime parts. A338315 (A337987) does not consider singletons coprime. A338317 (A338316) gives these partitions. A337462 (A333227) gives pairwise coprime compositions. A337485 (A337984) gives pairwise coprime integer partitions with no 1's. A337665 (A333228) gives compositions with pairwise coprime distinct parts. A337667 (A337666) gives pairwise non-coprime compositions. A337697 (A022340 /\ A333227) = pairwise coprime compositions with no 1's. A337983 (A337696) gives pairwise non-coprime strict compositions, with unordered version A318717 (A318719). Cf. A051424, A289508, A302569, A303140, A337694. Sequence in context: A066639 A370808 A111212 * A141286 A165686 A025209 Adjacent sequences: A338314 A338315 A338316 * A338318 A338319 A338320 KEYWORD nonn AUTHOR Gus Wiseman, Oct 24 2020 STATUS approved

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Last modified September 12 21:40 EDT 2024. Contains 375855 sequences. (Running on oeis4.)