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 A128039 #10 Feb 16 2025 08:33:04
%S A128039 3,6,10,13,18,26,29,218,220,223,491,535,538,622,628,3121,3126,3148,
%T A128039 3150,3155,3159,4348,4436,4440,4444,4458,4476,4485,4506,4556,4608,
%U A128039 4611,4761,5066,5783,5788,12528,1061290,2785126,2785691,2867466,2867469,2872437
%N A128039 Numbers n such that 1 - Sum_{k=1..n-1} A001223(k)*(-1)^k = 0.
%C A128039 Sequence has 294 terms < 10^7. If n is in this sequence then prime(n) = abs(3 + 2*Sum_{k=1..n-1} prime(k)*(-1)^k).
%H A128039 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeDifferenceFunction.html">Prime Difference Function</a>
%H A128039 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeSums.html">Prime Sums</a>
%H A128039 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlternatingSeries.html">Alternating Series</a>
%e A128039 1 - ( -A001223(1) + A001223(2)) = 1-(-1+2) = 0, hence 3 is a term.
%e A128039 1 - ( -A001223(1) + A001223(2) - A001223(3) + A001223(4) - A001223(5)) = 1-(-1+2-2+4-2) = 0, hence 6 is a term.
%t A128039 S=0; a=0; Do[S=S+((Prime[k+1]-Prime[k])*(-1)^k); If[1-S==0, a++; Print[a," ",k+1]], {k, 1, 10^7, 1}]
%Y A128039 Cf. A127596, A001223 (differences between consecutive primes), A000101 (increasing gaps between primes, upper end), A002386 (increasing gaps between primes, lower end), A066033.
%K A128039 nonn
%O A128039 1,1
%A A128039 _Manuel Valdivia_, May 07 2007