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A000741
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Number of compositions of n into 3 ordered relatively prime parts.
(Formerly M2531 N0999)
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19
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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
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OFFSET
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1,4
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REFERENCES
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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).
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LINKS
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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
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FORMULA
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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
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EXAMPLE
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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)
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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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.
A291166 intersected with A014311 ranks these compositions.
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.
Cf. A001399, A007997, A023022, A078374, A337450, A337451, A337602.
Sequence in context: A133331 A276381 A259728 * A133205 A049991 A143981
Adjacent sequences: A000738 A000739 A000740 * A000742 A000743 A000744
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Edited by Alois P. Heinz, Feb 08 2011
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STATUS
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approved
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