A024447 - OEIS

%I #38 Aug 05 2024 06:27:52

%S 0,6,31,101,288,652,1349,2451,4222,7122,11121,17041,25118,35352,48559,

%T 65943,88422,115262,148829,189157,235804,292052,357705,435491,528902,

%U 635962,755545,890793,1040232,1207472,1409783,1635103,1888690,2165022,2481945

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

%C a(n) is the 2nd elementary symmetric function of the first n+1 primes.

%C 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

%H Robert Israel, <a href="/A024447/b024447.txt">Table of n, a(n) for n = 1..10000</a>

%F a(1) = 0, a(n+1) = prime(n+1)*(sum of first n primes) + a(n), for n > 1.

%F a(n) = ((A007504(n))^2 - A024450(n))/2. - _Timothy Varghese_, May 06 2014

%F a(n) ~ (3*n^4*log^2(n) - 4*n^3*log^2(n))/24. - _Timothy Varghese_, May 06 2014

%p Primes:= [seq](ithprime(i),i=1..100):

%p (map(`^`,ListTools:-PartialSums(Primes),2) - ListTools:-PartialSums(map(`^`,Primes,2)))/2; # _Robert Israel_, Sep 24 2015

%t a[1] = 0; a[n_] := a[n] = a[n-1] + Prime[n]*Total[Prime[Range[n-1]]];

%t Array[a, 35] (* _Jean-François Alcover_, Feb 28 2019 *)

%o (PARI) /* Extra memory allocation could be required. */

%o Primes=List();

%o forprime(x=2,prime(500000),listput(Primes,x));

%o /* Keep previous lines global, before a(n) */

%o 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

%Y Cf. A007504, A024448, A024449, A024450.

%K nonn

%O 1,2

%A _Clark Kimberling_