This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A116994 #15 Jan 23 2025 23:31:38
%S A116994 3,1759,3323,469303,605113,641969,1110587,1426669,11148289,18352349,
%T A116994 20473721,21820391,24710753,30048589,36690923,40785301,97060681,
%U A116994 155135369,160593239,168132247,361391623,377965069,416572171,645803201
%N A116994 Prime partial sums of triangular numbers with prime indices.
%H A116994 Harvey P. Dale, <a href="/A116994/b116994.txt">Table of n, a(n) for n = 1..2000</a>
%F A116994 A000040 INTERSECTION A085739. Primes in A085739.
%e A116994 a(1) = Sum_{i=1..1} prime(i)*(prime(i)+1)/2 = T(2) = 3.
%e A116994 a(2) = Sum_{i=1..11} prime(i)*(prime(i)+1)/2 = T(2)+T(3)+T(5)+T(7)+T(11)+T(13)+T(17)+T(19)+T(23)+T(29)+T(31) = 1759.
%e A116994 a(3) = Sum_{i=1..13} prime(i)*(prime(i)+1)/2 = 3323.
%e A116994 a(4) = Sum_{i=1..53} prime(i)*(prime(i)+1)/2 = T(2) + ... + T(241) = 469303.
%e A116994 a(5) = Sum_{i=1..57} prime(i)*(prime(i)+1)/2 = T(2) + ... + T(269) = 605113.
%e A116994 a(6) = Sum_{i=1..58} prime(i)*(prime(i)+1)/2 = T(2) + ... + T(271) = 641969.
%e A116994 a(7) = Sum_{i=1..68} prime(i)*(prime(i)+1)/2 = T(2) + ... + T(337) = 1110587.
%p A116994 T:=n->n*(n+1)/2: a:=proc(n): if isprime(sum(T(ithprime(j)),j=1..n))=true then sum(T(ithprime(j)),j=1..n) else fi end: seq(a(n),n=1..500); # _Emeric Deutsch_, Apr 06 2006
%t A116994 Select[Accumulate[Table[(n(n+1))/2,{n,Prime[Range[500]]}]],PrimeQ] (* _Harvey P. Dale_, Jan 25 2015 *)
%Y A116994 Cf. A000040, A000217, A034953, A085739.
%K A116994 easy,nonn
%O A116994 1,1
%A A116994 _Jonathan Vos Post_, Apr 02 2006
%E A116994 More terms from _Emeric Deutsch_, Apr 06 2006