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 A172112 #5 Oct 09 2023 15:15:02
%S A172112 3,10,23,42,79,122,189,268,365,468,577,704,867,1060,1283,1512,1789,
%T A172112 2096,2409,2758,3137,3534,3973,4430,4893,5380,5879,6492,7135,7808,
%U A172112 8547,9304,10073,10896,11749,12608,13485,14368,15275,16212,17179,18188,19275
%N A172112 Partial sums of A023200.
%C A172112 Primes in the partial sum begin: a(1) = 3, a(3) = 23, a(5) = 79, a(11) = 577, a(15) = 1283, a(17) = 1789, a(21) = 3137, a(27) = 5879. Of these, the smaller members of cousin prime pairs which appear among the partial sums of smaller member p of cousin prime pairs begin: 3, 79; which are the next in this subset?
%F A172112 a(n) = SUM[i=i..n] A023200(i) = SUM[i=i..n] {Primes p such that p and p + 4 are both primes}.
%e A172112 a(30) = 3 + 7 + 13 + 19 + 37 + 43 + 67 + 79 + 97 + 103 + 109 + 127 + 163 + 193 + 223 + 229 + 277 + 307 + 313 + 349 + 379 + 397 + 439 + 457 + 463 + 487 + 499 + 613 + 643 + 673 = 7808.
%t A172112 Accumulate[Select[Prime[Range[250]],PrimeQ[#+4]&]] (* _Harvey P. Dale_, Oct 09 2023 *)
%Y A172112 Cf. A000040, A023200, A046132.
%K A172112 easy,nonn
%O A172112 1,1
%A A172112 _Jonathan Vos Post_, Jan 25 2010
%E A172112 More terms from _Max Alekseyev_, Jan 31 2010