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A062022 a(n) = Sum_{k=1..n} Sum_{j=1..k} (prime(k) - prime(j))^2.

%I A062022 #13 May 05 2022 04:41:52
%S A062022 0,1,14,59,256,581,1298,2287,4004,7329,11338,17915,26660,36637,49406,
%T A062022 67239,91252,117585,151730,191819,235112,289013,350842,425919,521300,
%U A062022 628001,740666,865899,997744,1143501,1345454,1565639,1815068,2074761
%N A062022 a(n) = Sum_{k=1..n} Sum_{j=1..k} (prime(k) - prime(j))^2.
%H A062022 G. C. Greubel, <a href="/A062022/b062022.txt">Table of n, a(n) for n = 1..1000</a>
%F A062022 From _G. C. Greubel_, May 04 2022: (Start)
%F A062022 a(n) = a(n-1) + n*prime(n)^2 + Sum_{k=1..n} prime(k)*(prime(k) - 2*prime(n)), with a(0) = a(1) = 0.
%F A062022 a(n) = n*Sum_{j=1..n} prime(j)^2 - (Sum_{j=1..n} prime(j))^2 = n*A024450(n) - A007504(n)^2. (End)
%e A062022 a(3) = (5-2)^2 + (5-3)^2 + (3-2)^2 = 14, sum of the squared differences of all pairs of the first 3 primes.
%p A062022 A062022 := proc(n)
%p A062022     local a,i,j ;
%p A062022     a := 0 ;
%p A062022     for j from 1 to n do
%p A062022         for i from 1 to j-1 do
%p A062022             a := a+(ithprime(j)-ithprime(i))^2 ;
%p A062022         end do:
%p A062022     end do:
%p A062022     a ;
%p A062022 end proc:
%p A062022 seq(A062022(n), n=1..10); # _R. J. Mathar_, Oct 03 2014
%t A062022 a[n_]:= a[n]= n*Sum[Prime[k]^2, {k,n}] - (Sum[Prime[j], {j,n}])^2;
%t A062022 Table[a[n], {n, 50}] (* _G. C. Greubel_, May 04 2022 *)
%o A062022 (SageMath)
%o A062022 @CachedFunction
%o A062022 def a(n): return n*sum(nth_prime(j)^2 for j in (1..n)) - (sum(nth_prime(j) for j in (1..n)))^2
%o A062022 [a(n) for n in (1..50)] # _G. C. Greubel_, May 04 2022
%Y A062022 Cf. A000040, A007504, A024450, A062020, A062021.
%K A062022 nonn
%O A062022 1,3
%A A062022 _Amarnath Murthy_, Jun 02 2001
%E A062022 More terms from _Matthew Conroy_, Jun 11 2001