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A024447 Sum of the products of the primes taken 2 at a time from first n primes. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) = 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 series. - 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[n-1] + Prime[n]*Total[Prime[Range[n-1]]];

Array[a, 35] (* Jean-Fran├žois 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 A166786

Adjacent sequences:  A024444 A024445 A024446 * A024448 A024449 A024450

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified June 18 18:52 EDT 2019. Contains 324215 sequences. (Running on oeis4.)