A062020 a(n) = Sum_{i=1..n} Sum_{j=1..i} (prime(i) - prime(j)).

0, 1, 6, 17, 44, 81, 142, 217, 324, 485, 666, 913, 1208, 1529, 1906, 2373, 2936, 3533, 4238, 5019, 5840, 6787, 7822, 8995, 10360, 11825, 13342, 14967, 16648, 18445, 20662, 23003, 25536, 28135, 31074, 34083, 37308, 40755, 44354, 48187, 52260

Offset: 1

Author

Amarnath Murthy, Jun 02 2001

Keywords

• nonn

Examples

			a(3) = (5-2) + (5-3) + (3-2) = 6.
		

Links

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

Crossrefs

Cf. A000040, A007504, A014285, A062021, A062022.

Programs

Mathematica

a[n_]:= a[n]= If[n<3, (n-1), 2*a[n-1] -a[n-2] +(n-1)*(Prime[n] -Prime[n-1])];
Table[a[n], {n, 50}] (* G. C. Greubel, May 04 2022 *)

SageMath

@CachedFunction
def a(n): # A062020
    if (n<3): return (n-1)
    else: return 2*a(n-1) - a(n-2) + (n-1)*(nth_prime(n) - nth_prime(n-1))
[a(n) for n in (1..50)] # G. C. Greubel, May 04 2022

Formula

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

a(n) = 2*a(n-1) - a(n-2) + (n-1)*(prime(n) - prime(n-1)), with a(1) = 0, a(2) = 1.

a(n) = Sum_{j=1..n} (2*j - (n+1))*prime(j) = 2*A014285(n) - (n+1)*A007504(n). - G. C. Greubel, May 04 2022

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jun 05 2001

Name edited by G. C. Greubel, May 04 2022