A062022 a(n) = Sum_{k=1..n} Sum_{j=1..k} (prime(k) - prime(j))^2.

0, 1, 14, 59, 256, 581, 1298, 2287, 4004, 7329, 11338, 17915, 26660, 36637, 49406, 67239, 91252, 117585, 151730, 191819, 235112, 289013, 350842, 425919, 521300, 628001, 740666, 865899, 997744, 1143501, 1345454, 1565639, 1815068, 2074761

Offset: 1

Author

Amarnath Murthy, Jun 02 2001

Keywords

• nonn

Examples

			a(3) = (5-2)^2 + (5-3)^2 + (3-2)^2 = 14, sum of the squared differences of all pairs of the first 3 primes.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040, A007504, A024450, A062020, A062021.

Programs

Maple

A062022 := proc(n)
    local a,i,j ;
    a := 0 ;
    for j from 1 to n do
        for i from 1 to j-1 do
            a := a+(ithprime(j)-ithprime(i))^2 ;
        end do:
    end do:
    a ;
end proc:
seq(A062022(n), n=1..10); # R. J. Mathar, Oct 03 2014

Mathematica

a[n_]:= a[n]= n*Sum[Prime[k]^2, {k,n}] - (Sum[Prime[j], {j,n}])^2;
Table[a[n], {n, 50}] (* G. C. Greubel, May 04 2022 *)

SageMath

@CachedFunction
def a(n): return n*sum(nth_prime(j)^2 for j in (1..n)) - (sum(nth_prime(j) for j in (1..n)))^2
[a(n) for n in (1..50)] # G. C. Greubel, May 04 2022

Formula

From G. C. Greubel, May 04 2022: (Start)

a(n) = a(n-1) + n*prime(n)^2 + Sum_{k=1..n} prime(k)*(prime(k) - 2*prime(n)), with a(0) = a(1) = 0.

a(n) = n*Sum_{j=1..n} prime(j)^2 - (Sum_{j=1..n} prime(j))^2 = n*A024450(n) - A007504(n)^2. (End)

Extensions

More terms from Matthew Conroy, Jun 11 2001