

A024447


Sum of the products of the primes taken 2 at a time from the first n primes.


9



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

1,2


COMMENTS

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


LINKS

Robert Israel, Table of n, a(n) for n = 1..10000


FORMULA

a(1) = 0, a(n+1) = prime(n+1)*(sum of first n primes) + a(n), for n > 1.
a(n) = ((A007504(n))^2  A024450(n)))/2.  Timothy Varghese, May 06 2014
a(n) ~ (3*n^4*log^2(n)  4*n^3*log^2(n))/24.  Timothy Varghese, May 06 2014


MAPLE

Primes:= [seq](ithprime(i), i=1..100):
(map(`^`, ListTools:PartialSums(Primes), 2)  ListTools:PartialSums(map(`^`, Primes, 2)))/2; # Robert Israel, Sep 24 2015


MATHEMATICA

a[1] = 0; a[n_] := a[n] = a[n1] + Prime[n]*Total[Prime[Range[n1]]];
Array[a, 35] (* JeanFrançois Alcover, Feb 28 2019 *)


PROG

(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


CROSSREFS

Cf. A007504, A024448, A024449, A024450.
Sequence in context: A096959 A112562 A244716 * A303172 A143568 A337574
Adjacent sequences: A024444 A024445 A024446 * A024448 A024449 A024450


KEYWORD

nonn


AUTHOR

Clark Kimberling


STATUS

approved



