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a(n) = Sum_{i=n..n+3}Sum_{j=i+1..n+4}prime(i)*prime(j).
5

%I #5 Mar 31 2012 10:22:03

%S 288,574,1078,1750,2710,4006,5590,7630,10270,13030,15766,19462,23510,

%T 27550,32830,38590,43750,49190,55570,62302,70726,80470,89350,98710,

%U 106870,113590,124822,137590,151990,167230,186454,199798,214774,230270

%N a(n) = Sum_{i=n..n+3}Sum_{j=i+1..n+4}prime(i)*prime(j).

%C a(n) = absolute value of the coefficient of x^3 of the polynomial Prod_{j=0,4}(x-prime(n+j)) of degree 5; the roots of this polynomial are prime(n), ..., prime(n+4); cf. Vieta's formulas.

%C All terms are even.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/VietasFormulas.html">Vieta's Formulas</a>

%t Table[Prime[x] Prime[x + 1] + Prime[x] Prime[x + 2] + Prime[x] Prime[x + 3] + Prime[x] Prime[x + 4] + Prime[x + 1] Prime[x + 2] + Prime[x + 1] Prime[x + 3] + Prime[x + 1] Prime[x + 4] + Prime[x + 2] Prime[x + 3] + Prime[x + 2] Prime[x + 4] + Prime[x + 3] Prime[x + 4], {x, 1, 100}]

%o (PARI) 1. {m=34;k=4;for(n=1,m,print1(sum(i=n,n+k-1,sum(j=i+1,n+k,prime(i)*prime(j))),","))} 2. {m=34;k=4;for(n=1,m,print1(abs(polcoeff(prod(j=0,k,(x-prime(n+j))),3)),","))} - Klaus Brockhaus, Jan 21 2007

%Y Cf. A127345, A127346, A127347, A127348, A127349, A127351, A070934, A006094.

%K nonn

%O 1,1

%A _Artur Jasinski_, Jan 11 2007

%E Edited by _Klaus Brockhaus_, Jan 21 2007