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A024447
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Sum of the products of the primes taken 2 at a time from the first n primes.
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10
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0, 6, 31, 101, 288, 652, 1349, 2451, 4222, 7122, 11121, 17041, 25118, 35352, 48559, 65943, 88422, 115262, 148829, 189157, 235804, 292052, 357705, 435491, 528902, 635962, 755545, 890793, 1040232, 1207472, 1409783, 1635103, 1888690, 2165022, 2481945
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OFFSET
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1,2
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COMMENTS
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a(n) is the 2nd elementary symmetric function of the first n+1 primes.
Using the identity that (x_1 + x_2 + ... + x_n)^2 - (x_1^2 + x_ 2^2 + ... + x_n^2) is the sum of the products taken two at a time, a(n) can be expressed with the sum of the primes and the sum of the prime squared. Since they both have asymptotic formulas, this yields an asymptotic formula for this sequence. - Timothy Varghese, May 06 2014
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LINKS
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FORMULA
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a(1) = 0, a(n+1) = prime(n+1)*(sum of first n primes) + a(n), for n > 1.
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MAPLE
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Primes:= [seq](ithprime(i), i=1..100):
(map(`^`, ListTools:-PartialSums(Primes), 2) - ListTools:-PartialSums(map(`^`, Primes, 2)))/2; # Robert Israel, Sep 24 2015
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MATHEMATICA
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a[1] = 0; a[n_] := a[n] = a[n-1] + Prime[n]*Total[Prime[Range[n-1]]];
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PROG
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(PARI) /* Extra memory allocation could be required. */
Primes=List();
forprime(x=2, prime(500000), listput(Primes, x));
/* Keep previous lines global, before a(n) */
a(n)={my(p=vector(n, j, Primes[j]), s=0); forvec(y=vector(2, i, [1, #p]), s+=(p[y[1]]*p[y[2]]), 2); s} \\ R. J. Cano, Oct 11 2015
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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