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%I #13 Jan 18 2021 02:41:33
%S 0,0,0,1,1,2,2,3,4,4,5,6,8,7,10,7,11,11,17,12,19,12,19,17,29,16,28,19,
%T 31,23,46,23,42,25,45,27,59,31,57,34,61,37,84,38,75,42,74,47,107,45,
%U 98,51,96,56,135,54,115,63,117,67,174,65,139,75,144,75,194
%N Number of unordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is not considered coprime unless it is (1).
%C First differs from A337600 at a(9) = 4, A337600(9) = 5.
%H Fausto A. C. Cariboni, <a href="/A337601/b337601.txt">Table of n, a(n) for n = 0..10000</a>
%F For n > 0, a(n) = A337600(n) - A079978(n).
%e The a(3) = 1 through a(14) = 10 partitions (A = 10, B = 11, C = 12):
%e 111 211 221 321 322 332 441 433 443 543 544 554
%e 311 411 331 431 522 532 533 552 553 743
%e 511 521 531 541 551 651 661 752
%e 611 711 721 722 732 733 761
%e 811 731 741 751 833
%e 911 831 922 851
%e 921 B11 941
%e A11 A31
%e B21
%e C11
%t Table[Length[Select[IntegerPartitions[n,{3}],CoprimeQ@@Union[#]&]],{n,0,100}]
%Y A014612 intersected with A304711 ranks these partitions.
%Y A220377 is the strict case.
%Y A304709 counts these partitions of any length.
%Y A307719 is the strict case except for any number of 1's.
%Y A337600 considers singletons to be coprime.
%Y A337603 is the ordered version.
%Y A000217 counts 3-part compositions.
%Y A000837 counts relatively prime partitions.
%Y A001399/A069905/A211540 count 3-part partitions.
%Y A023023 counts relatively prime 3-part partitions.
%Y A051424 counts pairwise coprime or singleton partitions.
%Y A101268 counts pairwise coprime or singleton compositions.
%Y A305713 counts pairwise coprime strict partitions.
%Y A327516 counts pairwise coprime partitions.
%Y A333227 ranks pairwise coprime compositions.
%Y A333228 ranks compositions whose distinct parts are pairwise coprime.
%Y A337461 counts pairwise coprime 3-part compositions.
%Y Cf. A001840, A007359, A007360, A087087, A284825, A302696, A304712, A328673, A335238, A337602.
%K nonn
%O 0,6
%A _Gus Wiseman_, Sep 20 2020
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