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%I M0202 N0075 #25 Mar 09 2020 20:19:12
%S 1,0,2,2,1,4,1,4,2,2,3,2,2,4,3,2,4,2,4,4,4,2,5,2,6,2,5,0,4,2,6,4,4,2,
%T 7,0,8,2,3,2,6,2,8,4,6,2,7,2,10,2,8,0,6,2,10,2,6,0,7,2,12,4,5,2,10,0,
%U 12,2,4,2,10,2,12,4,9,2,10,0,14,2,8,2,9,2,16,2,9,0,8,2,18,2,8,0,9,0,14
%N Number of solutions to n=p+q where p and q are primes or zero.
%C Arises in studying the Goldbach conjecture.
%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 N_{n,2}]
%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 T. D. Noe, <a href="/A002126/b002126.txt">Table of n, a(n) for n = 0..10000</a>
%H P. A. MacMahon, <a href="http://plms.oxfordjournals.org/content/s2-23/1/290.extract">Properties of prime numbers deduced from the calculus of symmetric functions</a>, Proc. London Math. Soc., 23 (1923), 290-316. = Coll. Papers, II, pp. 354-380.
%F G.f.: (1 + Sum_i x^prime(i))^2. [Corrected by _T. D. Noe_, Dec 05 2006]
%o (PARI) (a(n) = sum(k=0, n, zp(k)*zp(n-k))); {zp(n) = if( n==0, 1, isprime(n))}; /* _Michael Somos_, Jul 26 1999 */
%Y Cf. A002375, A045917, A061358, A073610
%K nonn
%O 0,3
%A _N. J. A. Sloane_
%E a(54) corrected by _Paul Zimmermann_, Mar 15 1996
%E Better description from _Michael Somos_, Jul 26 1999
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