A357251 a(n) = Sum_{1<=i<=j<=n} prime(i)*prime(j).

4, 19, 69, 188, 496, 1029, 2015, 3478, 5778, 9519, 14479, 21768, 31526, 43609, 59025, 79218, 105178, 135739, 173795, 219164, 271140, 333629, 406171, 491878, 594698, 711959, 842151, 988848, 1150168, 1330177, 1548617, 1791098, 2063454, 2359107, 2698231, 3064708, 3470396, 3918157, 4404795, 4938846

Offset: 1

Author

J. M. Bergot and Robert Israel, Sep 20 2022

Keywords

• nonn

Comments

a(n) is the sum of products of unordered pairs of (not necessarily distinct) elements from the first n primes.

It appears that 4 is the only square in the sequence.

Examples

			a(3) = 2*2 + 2*3 + 2*5 + 3*3 + 3*5 + 5*5 = 69.
		

Links

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

Crossrefs

Cf. A007504, A024447, A024450, A065762, A357252.

Partial sums of A143215.

Row n=2 of A343751.

Programs

Maple

P:= [seq(ithprime(i),i=1..100)]:
S:= ListTools:-PartialSums(P):
ListTools:-PartialSums(zip(`*`,P,S));

Mathematica

Accumulate[(p = Prime[Range[40]]) * Accumulate[p]] (* Amiram Eldar, Sep 20 2022 *)

Python

from itertools import accumulate
from sympy import prime, primerange
def aupton(nn):
    p = list(primerange(2, prime(nn)+1))
    return list(accumulate(c*d for c, d in zip(p, accumulate(p))))
print(aupton(40)) # Michael S. Branicky, Sep 24 2022 after Amiram Eldar

Formula

a(n) = (A007504(n)^2 + A024450(n))/2.

a(n) = A024447(n) + A024450(n).

a(n) = A065762(n)/2. - Hugo Pfoertner, Sep 24 2022