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 A112632 #33 Nov 19 2023 21:16:56
%S A112632 1,1,2,1,2,1,2,1,2,3,2,1,2,1,2,3,4,3,2,3,2,1,2,3,2,3,2,3,2,3,2,3,4,3,
%T A112632 4,3,2,1,2,3,4,3,4,3,4,3,2,1,2,1,2,3,2,3,4,5,6,5,4,5,4,5,4,5,4,5,4,3,
%U A112632 4,3,4,5,4,3,2,3,4,3,4,3,4,3,4,3,2,3,4,3,4,3,4,5,4,5,4,5,6,7,6,5
%N A112632 Excess of 3k - 1 primes over 3k + 1 primes, beginning with 2.
%C A112632 Cumulative sums of A134323, negated. The first negative term is a(23338590792) = -1 for the prime 608981813029. See page 4 of the paper by Granville and Martin. - _T. D. Noe_, Jan 23 2008 [Corrected by _Jianing Song_, Nov 24 2018]
%C A112632 See the comment about "Chebyshev's bias" in A321856. - _Jianing Song_, Nov 24 2018
%H A112632 T. D. Noe, <a href="/A112632/b112632.txt">Table of n, a(n) for n = 1..10000</a>
%H A112632 A. Granville and G. Martin, <a href="http://www.jstor.org/stable/27641834">Prime number races</a>, Amer. Math. Monthly, 113 (No. 1, 2006), pp. 1-33.
%H A112632 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev%27s_bias">Chebyshev's bias</a>
%F A112632 a(n) = -Sum_{primes p<=n} Legendre(prime(i),3) = -Sum_{primes p<=n} Kronecker(-3,prime(i)) = -Sum_{i=1..n} A102283(prime(i)). - _Jianing Song_, Nov 24 2018
%e A112632 a(1) = 1 because 2 == -1 (mod 3).
%e A112632 a(2) = 1 because 3 == 0 (mod 3) and does not change the counting.
%e A112632 a(3) = 2 because 5 == -1 (mod 3).
%e A112632 a(4) = 1 because 7 == 1 (mod 3).
%t A112632 a[n_] := a[n] = a[n-1] + If[Mod[Prime[n], 6] == 1, -1, 1]; a[1] = a[2] = 1; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Jul 24 2012 *)
%t A112632 Accumulate[Which[IntegerQ[(#+1)/3],1,IntegerQ[(#-1)/3],-1,True,0]& /@ Prime[ Range[100]]] (* _Harvey P. Dale_, Jun 06 2013 *)
%o A112632 (Haskell)
%o A112632 a112632 n = a112632_list !! (n-1)
%o A112632 a112632_list = scanl1 (+) $ map negate a134323_list
%o A112632 -- _Reinhard Zumkeller_, Sep 16 2014
%o A112632 (PARI) a(n) = -sum(i=1, n, kronecker(-3, prime(i))) \\ _Jianing Song_, Nov 24 2018
%Y A112632 Cf. A007352, A098044, A102283, A134323.
%Y A112632 Let d be a fundamental discriminant.
%Y A112632 Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: A321860 (d=-11), A320857 (d=-8), A321859 (d=-7), A066520 (d=-4), A321856 (d=-3), A321857 (d=5), A071838 (d=8), A321858 (d=12).
%Y A112632 Sequences of the form "a(n) = -Sum_{i=1..n} Kronecker(d,prime(i))" with |d| <= 12: A321865 (d=-11), A320858 (d=-8), A321864 (d=-7), A038698 (d=-4), this sequence (d=-3), A321862 (d=5), A321861 (d=8), A321863 (d=12).
%K A112632 sign,nice
%O A112632 1,3
%A A112632 _Roger Hui_, Dec 22 2005