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%I A002120 M0414 N0158 #17 Oct 14 2023 22:34:34
%S A002120 0,-2,3,2,0,1,7,2,-6,8,22,-7,0,33,3,-14,51,46,-19,12,94,42,-23,113,
%T A002120 150,-54,48,345,116,-109,403,498,-140,219,1057,326,-259,1271,1641,
%U A002120 -308,656,3396,1161,-790,4269,5357,-987,2257,10934,3958,-1986,13678,17278,-2492,7447,35569,13778,-5860,44368,56403,-6405
%N A002120 a(1) = 0, a(2) = -2; for n > 2, a(n) + a(n-2) - a(n-3) - a(n-5) - ... - a(n-p) = (-1)^(n+1)*n if n is prime, otherwise = 0, where p = largest prime < n.
%C A002120 Arises in studying the Goldbach conjecture.
%D A002120 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 e_n]
%D A002120 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002120 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002120 N. J. A. Sloane, <a href="/A002120/b002120.txt">Table of n, a(n) for n = 1..1000</a>
%H A002120 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 A002120 a(n) = (-1)^(n+1)*n*A010051(n)+Sum_{k=1..n-1} (-1)^(n-k+1)*A010051(n-k)*a(k). - _Vladeta Jovovic_, May 08 2003
%p A002120 M:=90; e:=array(0..M); e[1]:=0; e[2]:=-2; for n from 3 to M do t1:=-e[n-2]; if isprime(n) then t1:=t1+(-1)^(n+1)*n; fi; for k from 2 to n do p := ithprime(k); if p < n then t1 := t1 + e[n-p]; fi; od: e[n]:=t1; od: [seq(e[n],n=1..M)];
%K A002120 sign
%O A002120 1,2
%A A002120 _N. J. A. Sloane_
%E A002120 More terms from _Vladeta Jovovic_, May 08 2003
%E A002120 Edited by _N. J. A. Sloane_, Dec 03 2006

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