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

1,2

Links

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

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.

Example

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A000040, A062020, A062022.

Sequence in context: A222474 A327269 A266021 * A241780 A215785 A082145

Adjacent sequences: A062018 A062019 A062020 * A062022 A062023 A062024

Keyword

nonn

Author

Amarnath Murthy, Jun 02 2001

Extensions

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

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

Status

approved