cp's OEIS Frontend

A101268 Number of compositions of n into pairwise relatively prime parts.

%I A101268 #28 Jan 05 2025 19:51:38
%S A101268 1,1,2,4,7,13,22,38,63,101,160,254,403,635,984,1492,2225,3281,4814,
%T A101268 7044,10271,14889,21416,30586,43401,61205,85748,119296,164835,226423,
%U A101268 309664,422302,574827,781237,1060182,1436368,1942589,2622079,3531152,4742316,6348411
%N A101268 Number of compositions of n into pairwise relatively prime parts.
%C A101268 Here a singleton is always considered pairwise relatively prime. Compare to A337462. - _Gus Wiseman_, Oct 18 2020
%H A101268 Fausto A. C. Cariboni, <a href="/A101268/b101268.txt">Table of n, a(n) for n = 0..500</a> (terms 0..400 from Alois P. Heinz)
%H A101268 Temba Shonhiwa, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/44-4/quarttemba04_2006.pdf">Compositions with pairwise relatively prime summands within a restricted setting</a>, Fibonacci Quart. 44 (2006), no. 4, 316-323.
%F A101268 It seems that no formula is known.
%e A101268 From _Gus Wiseman_, Oct 18 2020: (Start)
%e A101268 The a(1) = 1 through a(5) = 13 compositions:
%e A101268   (1)  (2)   (3)    (4)     (5)
%e A101268        (11)  (12)   (13)    (14)
%e A101268              (21)   (31)    (23)
%e A101268              (111)  (112)   (32)
%e A101268                     (121)   (41)
%e A101268                     (211)   (113)
%e A101268                     (1111)  (131)
%e A101268                             (311)
%e A101268                             (1112)
%e A101268                             (1121)
%e A101268                             (1211)
%e A101268                             (2111)
%e A101268                             (11111)
%e A101268 (End)
%t A101268 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[#]<=1||CoprimeQ@@#&]],{n,0,10}] (* _Gus Wiseman_, Oct 18 2020 *)
%Y A101268 Row sums of A282748.
%Y A101268 A051424 is the unordered version, with strict case A007360.
%Y A101268 A335235 ranks these compositions.
%Y A101268 A337461 counts these compositions of length 3, with unordered version A307719 and unordered strict version A220377.
%Y A101268 A337462 does not consider a singleton to be coprime unless it is (1), with strict version A337561.
%Y A101268 A337562 is the strict case.
%Y A101268 A337664 looks only at distinct parts, with non-constant version A337665.
%Y A101268 A000740 counts relatively prime compositions, with strict case A332004.
%Y A101268 A178472 counts compositions with a common factor.
%Y A101268 Cf. A087087, A302569, A305713, A326675, A327516, A328673, A333227, A333228.
%K A101268 nonn
%O A101268 0,3
%A A101268 _Vladeta Jovovic_, Dec 18 2004
%E A101268 a(0)=1 prepended by _Alois P. Heinz_, Jun 14 2017