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%I M0154 N0062
%S 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,
%T 220,277,373,476,616,810,1037,1361,1763,2279,2984,3846,5006,6521,8428,
%U 10983,14249,18480,24048,31178,40520,52635,68281,88765,115211,149593,194381,252280,327696,425587,552527,717721
%N Number of compositions of n into a sum of odd primes.
%C Arises in studying the Goldbach conjecture.
%C 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 _Simon Plouffe_ in his 1992 dissertation is wrong.
%D 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]
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H N. J. A. Sloane, <a href="/A002124/b002124.txt">Table of n, a(n) for n = 0..1000</a>
%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 300
%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%F a(0)=1, a(1)=a(2)=0; for n >= 3, a(n) = Sum_{ primes p with 3 <= p <= n} a(n-p). [MacMahon]
%F G.f. 1/( 1 - sum(k>=2, x^A000040(k) ) ). [_Joerg Arndt_, Sep 30 2012]
%p A002124 := proc(n) coeff(series(1/(1-add(z^numtheory[ithprime](j),j=2..n)),z=0,n+1),z,n) end;
%p 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]
%t 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 *)
%Y Cf. A002125, A023360, A024939, A077608.
%K nonn
%O 0,9
%A _N. J. A. Sloane_.
%E Better description and more terms from _Philippe Flajolet_, Nov 11 2002
%E Edited by _N. J. A. Sloane_, Dec 03 2006
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