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

%I A357251 #21 Sep 29 2023 20:53:11
%S A357251 4,19,69,188,496,1029,2015,3478,5778,9519,14479,21768,31526,43609,
%T A357251 59025,79218,105178,135739,173795,219164,271140,333629,406171,491878,
%U A357251 594698,711959,842151,988848,1150168,1330177,1548617,1791098,2063454,2359107,2698231,3064708,3470396,3918157,4404795,4938846
%N A357251 a(n) = Sum_{1<=i<=j<=n} prime(i)*prime(j).
%C A357251 a(n) is the sum of products of unordered pairs of (not necessarily distinct) elements from the first n primes.
%C A357251 It appears that 4 is the only square in the sequence.
%H A357251 Robert Israel, <a href="/A357251/b357251.txt">Table of n, a(n) for n = 1..10000</a>
%F A357251 a(n) = (A007504(n)^2 + A024450(n))/2.
%F A357251 a(n) = A024447(n) + A024450(n).
%F A357251 a(n) = A065762(n)/2. - _Hugo Pfoertner_, Sep 24 2022
%e A357251 a(3) = 2*2 + 2*3 + 2*5 + 3*3 + 3*5 + 5*5 = 69.
%p A357251 P:= [seq(ithprime(i),i=1..100)]:
%p A357251 S:= ListTools:-PartialSums(P):
%p A357251 ListTools:-PartialSums(zip(`*`,P,S));
%t A357251 Accumulate[(p = Prime[Range[40]]) * Accumulate[p]] (* _Amiram Eldar_, Sep 20 2022 *)
%o A357251 (Python)
%o A357251 from itertools import accumulate
%o A357251 from sympy import prime, primerange
%o A357251 def aupton(nn):
%o A357251     p = list(primerange(2, prime(nn)+1))
%o A357251     return list(accumulate(c*d for c, d in zip(p, accumulate(p))))
%o A357251 print(aupton(40)) # _Michael S. Branicky_, Sep 24 2022 after _Amiram Eldar_
%Y A357251 Cf. A007504, A024447, A024450, A065762, A357252.
%Y A357251 Partial sums of A143215.
%Y A357251 Row n=2 of A343751.
%K A357251 nonn
%O A357251 1,1
%A A357251 _J. M. Bergot_ and _Robert Israel_, Sep 20 2022