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A023360
Number of compositions of n into prime parts.
44
1, 0, 1, 1, 1, 3, 2, 6, 6, 10, 16, 20, 35, 46, 72, 105, 152, 232, 332, 501, 732, 1081, 1604, 2352, 3493, 5136, 7595, 11212, 16534, 24442, 36039, 53243, 78573, 115989, 171264, 252754, 373214, 550863, 813251, 1200554, 1772207, 2616338, 3862121, 5701553
OFFSET
0,6
REFERENCES
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 292-295.
Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 501 terms from T. D. Noe)
S. R. Finch, Kalmar's composition constant, June 5, 2003. [Cached copy, with permission of the author]
Philippe Flajolet, More information including asymptotic form (1995). [Broken link]
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 43, 298
FORMULA
a(n) = Sum_{prime p<=n} a(n-p) with a(0)=1. - Henry Bottomley, Dec 15 2000
G.f.: 1/(1 - Sum_{k>=1} x^A000040(k)). - Andrew Howroyd, Dec 28 2017
EXAMPLE
2; 3; 4 = 2+2; 5 = 2+3 = 3+2; 6 = 2+2+2 = 3+3; 7 = 2+2+3 = 2+3+2 = 3+2+2 = 2+5 = 5+2; etc.
MAPLE
a:= proc(n) option remember; `if`(n=0, 1, add(
`if`(isprime(j), a(n-j), 0), j=1..n))
end:
seq(a(n), n=0..50); # Alois P. Heinz, Feb 12 2021
MATHEMATICA
CoefficientList[ Series[1 / (1 - Sum[ x^Prime[i], {i, 15}]), {x, 0, 45}], x]
PROG
(PARI) {my(n=60); Vec(1/(1-sum(k=1, n, if(isprime(k), x^k, 0))) + O(x*x^n))} \\ Andrew Howroyd, Dec 28 2017
CROSSREFS
Cf. A000607 for the unordered (partition) version.
Sequence in context: A347742 A286970 A347743 * A154028 A157793 A096375
KEYWORD
nonn
STATUS
approved