|
| |
|
|
A002124
|
|
Number of compositions of n into a sum of odd primes.
(Formerly M0154 N0062)
|
|
3
| |
|
|
1, 0, 0, 1, 0, 1, 1, 1, 2, 1, 3, 4, 3, 7, 7, 8, 14, 15, 21, 28, 33, 47, 58, 76, 103, 125, 169, 220, 277, 373, 476, 616, 810, 1037, 1361, 1763, 2279, 2984, 3846, 5006, 6521, 8428, 10983, 14249, 18480, 24048, 31178, 40520, 52635, 68281, 88765, 115211, 149593, 194381, 252280, 327696, 425587, 552527, 717721
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,9
|
|
|
COMMENTS
| Arises in studying the Goldbach conjecture.
The g.f. -(z-1)*(z+1)*(z**2+z+1)*(z**2-z+1)/(1-z**6-z**3-z**5-z**7+z**9) conjectured by S. Plouffe in his 1992 dissertation is wrong.
|
|
|
REFERENCES
| P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. [Coll. Papers, Vol. II, pp. 354-382] [The sequence i_n]
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
| N. J. A. Sloane, Table of n, a(n) for n = 0..1000
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 300
|
|
|
FORMULA
| a(0)=1, a(1)=a(2)=0; for n >= 3, a(n) = Sum_{ primes p with 3 <= p <= n} a(n-p). [MacMahon]
|
|
|
MAPLE
| A002124 := proc(n) coeff(series(1/(1-add(z^numtheory[ithprime](j), j=2..n)), z=0, n+1), z, n) end;
M:=120; a:=array(0..M); a[0]:=1; a[1]:=0; a[2]:=0; for n from 3 to M do t1:=0; for k from 2 to n do p := ithprime(k); if p <= n then t1 := t1 + a[n-p]; fi; od: a[n]:=t1; od: [seq(a[n], n=0..M)]; [N. J. A. Sloane, after MacMahon, Dec 03 2006] [Used in A002125]
|
|
|
MATHEMATICA
| a[0] = 1; a[1] = a[2] = 0; a[n_] := a[n] = (s = 0; p = 3; While[p <= n, s = s + a[n-p]; p = NextPrime[p]]; s); a /@ Range[0, 58] (* From Jean-François Alcover, Jun 28 2011, after P. A. MacMahon *)
|
|
|
CROSSREFS
| Cf. A002125, A023360, A024939, A077608.
Sequence in context: A144254 A133310 A077608 * A097564 A128270 A151550
Adjacent sequences: A002121 A002122 A002123 * A002125 A002126 A002127
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| Better description and more terms from Philippe Flajolet (Philippe.Flajolet(AT)inria.fr), Nov 11 2002
Edited by N. J. A. Sloane (njas(AT)research.att.com), Dec 03 2006
|
| |
|
|
