A062021 a(n) = Sum_{i=1..n} Sum_{j=1..i} (prime(i)^2 - prime(j)^2).

0, 5, 42, 151, 548, 1185, 2542, 4403, 7608, 13621, 20834, 32535, 47980, 65609, 88278, 119947, 162368, 208869, 269194, 340007, 416580, 512305, 622286, 756003, 925432, 1114661, 1314498, 1537015, 1771628, 2031993, 2393158, 2786315

Offset: 1

Author

Amarnath Murthy, Jun 02 2001

Keywords

• nonn

Examples

			a(3) = (5^2 - 2^2) + (5^2 - 3^2) + (3^2 - 2^2) = 42.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A062020, A062022.

Programs

Magma

[(&+[(&+[NthPrime(i)^2 - NthPrime(j)^2: j in [1..i]]): i in [1..n]]): n in [1..40]]; // G. C. Greubel, May 04 2022

Maple

N:= 100: # for a(1)..a(N)
P2:= [seq(ithprime(i)^2,i=1..N)]:
DP2:= P2[2..-1]-P2[1..-2]:
A[1]:= 0: A[2]:= 5:
for n from 3 to N do A[n]:= 2*A[n-1]+(n-1)*DP2[n-1]-A[n-2] od:
seq(A[i],i=1..N); # Robert Israel, Feb 02 2020

Mathematica

RecurrenceTable[{a[1]==0,a[2]==5,a[n]==2a[n-1]-a[n-2]+(n-1)(Prime[n]^2 - Prime[n-1]^2)}, a, {n,40}] (* Harvey P. Dale, May 16 2019 *)

SageMath

@CachedFunction
def a(n):
    if (n<3): return 5*(n-1)
    else: return 2*a(n-1) - a(n-2) + (n-1)*(nth_prime(n)^2 - nth_prime(n-1)^2)
[a(n) for n in (1..40)] # G. C. Greubel, May 04 2022

Formula

a(n) = 2*a(n-1) - a(n-2) + (n-1)*(prime(n)^2 - prime(n-1)^2) with a(1) = 0, a(2) = 5.

Extensions

More terms and formula from Larry Reeves (larryr(AT)acm.org), Jun 06 2001

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