cp's OEIS Frontend

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

%I A024447 #38 Aug 05 2024 06:27:52
%S A024447 0,6,31,101,288,652,1349,2451,4222,7122,11121,17041,25118,35352,48559,
%T A024447 65943,88422,115262,148829,189157,235804,292052,357705,435491,528902,
%U A024447 635962,755545,890793,1040232,1207472,1409783,1635103,1888690,2165022,2481945
%N A024447 Sum of the products of the primes taken 2 at a time from the first n primes.
%C A024447 a(n) is the 2nd elementary symmetric function of the first n+1 primes.
%C A024447 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 A024447 Robert Israel, <a href="/A024447/b024447.txt">Table of n, a(n) for n = 1..10000</a>
%F A024447 a(1) = 0, a(n+1) = prime(n+1)*(sum of first n primes) + a(n), for n > 1.
%F A024447 a(n) = ((A007504(n))^2 - A024450(n))/2. - _Timothy Varghese_, May 06 2014
%F A024447 a(n) ~ (3*n^4*log^2(n) - 4*n^3*log^2(n))/24. - _Timothy Varghese_, May 06 2014
%p A024447 Primes:= [seq](ithprime(i),i=1..100):
%p A024447 (map(`^`,ListTools:-PartialSums(Primes),2) - ListTools:-PartialSums(map(`^`,Primes,2)))/2; # _Robert Israel_, Sep 24 2015
%t A024447 a[1] = 0; a[n_] := a[n] = a[n-1] + Prime[n]*Total[Prime[Range[n-1]]];
%t A024447 Array[a, 35] (* _Jean-François Alcover_, Feb 28 2019 *)
%o A024447 (PARI) /* Extra memory allocation could be required. */
%o A024447 Primes=List();
%o A024447 forprime(x=2,prime(500000),listput(Primes,x));
%o A024447 /* Keep previous lines global, before a(n) */
%o A024447 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 A024447 Cf. A007504, A024448, A024449, A024450.
%K A024447 nonn
%O A024447 1,2
%A A024447 _Clark Kimberling_