A024447 Sum of the products of the primes taken 2 at a time from the first n primes.

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

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

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

Crossrefs

Cf. A007504, A024448, A024449, A024450.

Programs

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-François Alcover, Feb 28 2019 *)

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

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