A000741 Number of compositions of n into 3 ordered relatively prime parts.
0, 0, 1, 3, 6, 9, 15, 18, 27, 30, 45, 42, 66, 63, 84, 84, 120, 99, 153, 132, 174, 165, 231, 180, 270, 234, 297, 270, 378, 276, 435, 360, 450, 408, 540, 414, 630, 513, 636, 552, 780, 558, 861, 690, 828, 759, 1035, 744, 1113, 870, 1104, 972, 1326, 945, 1380, 1116, 1386, 1218
Offset: 1
Examples
From _Gus Wiseman_, Oct 14 2020: (Start)
The a(3) = 1 through a(8) = 18 triples:
(1,1,1) (1,1,2) (1,1,3) (1,1,4) (1,1,5) (1,1,6)
(1,2,1) (1,2,2) (1,2,3) (1,2,4) (1,2,5)
(2,1,1) (1,3,1) (1,3,2) (1,3,3) (1,3,4)
(2,1,2) (1,4,1) (1,4,2) (1,4,3)
(2,2,1) (2,1,3) (1,5,1) (1,5,2)
(3,1,1) (2,3,1) (2,1,4) (1,6,1)
(3,1,2) (2,2,3) (2,1,5)
(3,2,1) (2,3,2) (2,3,3)
(4,1,1) (2,4,1) (2,5,1)
(3,1,3) (3,1,4)
(3,2,2) (3,2,3)
(3,3,1) (3,3,2)
(4,1,2) (3,4,1)
(4,2,1) (4,1,3)
(5,1,1) (4,3,1)
(5,1,2)
(5,2,1)
(6,1,1)
(End)
References
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..10000
- H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2(4) (1964), 241-260.
- C. Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.
- N. J. A. Sloane, Transforms
Crossrefs
A000010 is the length-2 version.
A000217(n-2) does not require relative primality.
A000740 counts these compositions of any length.
A000742 is the length-4 version.
A000837 counts relatively prime partitions.
A023023 is the unordered version.
A101271 is the strict case.
A101391 has this as column k = 3.
A284825*6 is the pairwise non-coprime case.
A337461 is the pairwise coprime instead of relatively prime version.
A337603 counts length-3 compositions whose distinct parts are pairwise coprime.
A337604 is the pairwise non-coprime instead of relatively prime version.
Programs
-
Maple
with(numtheory): mobtr:= proc(p) proc(n) option remember; add(mobius(n/d)*p(d), d=divisors(n)) end end: A000217:= n-> n*(n+1)/2: a:= mobtr(n-> A000217(n-2)): seq(a(n), n=1..58); # Alois P. Heinz, Feb 08 2011 -
Mathematica
mobtr[p_] := Module[{f}, f[n_] := f[n] = Sum[MoebiusMu[n/d]*p[d], {d, Divisors[n]}]; f]; A000217[n_] := n*(n+1)/2; a = mobtr[A000217[#-2]&]; Table[a[n], {n, 1, 58}] (* Jean-François Alcover, Mar 12 2014, after Alois P. Heinz *) Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],GCD@@#==1&]],{n,0,30}] (* Gus Wiseman, Oct 14 2020 *)
Formula
Moebius transform of A000217(n-2).
G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = (1 - 3*x + 3*x^2)/(1 - x)^3. - Ilya Gutkovskiy, Apr 26 2017
Extensions
Edited by Alois P. Heinz, Feb 08 2011