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%I M0154 N0062 #47 Apr 13 2022 13:25:16
%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 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 P. A. MacMahon, <a href="https://doi.org/10.1112/plms/s2-23.1.290">Properties of prime numbers deduced from the calculus of symmetric functions</a>, Proc. London Math. Soc., 23 (1923), 290-316. [Coll. Papers, II, pp. 354-382] [The sequence i_n]
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%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] (* _Jean-François Alcover_, Jun 28 2011, after P. A. MacMahon *)
%o (Haskell)
%o import Data.List (genericIndex)
%o a002124 n = genericIndex a002124_list n
%o a002124_list = 1 : f 1 [] a065091_list where
%o f x qs ps'@(p:ps)
%o | p <= x = f x (p:qs) ps
%o | otherwise = sum (map (a002124 . (x -)) qs) : f (x + 1) qs ps'
%o -- _Reinhard Zumkeller_, Mar 21 2014
%Y Cf. A002125, A023360, A024939, A077608.
%Y Cf. A065091.
%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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