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 A178154 #8 Jun 07 2025 16:43:39
%S A178154 1,1,3,3,3,3,3,5,9,3,7,5,9,5,9,5,15,9,7,3,11,15,13,9,9,7,15,15,21,13,
%T A178154 11,9,9,19,9,15,19,9,9,17,27,21,15,15,7,21,25,7,27,9,21,15,9,27,15,21,
%U A178154 25,13,15,13,9,23,33,15,15,31,13,15,21,15,15,21,9,21,33,21,21,9,29,9,31,19
%N A178154 Sum_{a=1..(p-1)/2} LegendreSymbol(a, p) for primes p == 3 (mod 4).
%C A178154 Wenpeng Zhang proves that a(n) > 0. - _Peter Luschny_, Jun 07 2025
%D A178154 Richard Kenneth Guy, Unsolved Problems in Number Theory, Springer-Verlag, Berlin, 1994, pp. 244.
%H A178154 Wenpeng Zhang, <a href="https://doi.org/10.48550/arXiv.2506.02434">An old number theory problem related to the Legendre symbol</a>, arXiv:2506.02434 [math.HO], Jun. 2025.
%F A178154 A178153 for primes p=3 (mod 4).
%p A178154 P := upto -> select(n -> isprime(n) and modp(n, 4) = 3, [seq(1..upto)]):
%p A178154 a := p -> add(NumberTheory:-LegendreSymbol(a, p), a = 1..(p-1)/2):
%p A178154 seq(a(p), p in P(940)); # _Peter Luschny_, Jun 07 2025
%t A178154 Table[Length[Select[Range[(p-1)/2], JacobiSymbol[ #,p]==1&]] - Length[Select[Range[(p+1)/2,p-1], JacobiSymbol[ #,p]==1&]], {p, Select[Prime[Range[200]], Mod[ #,4]==3&]}]
%o A178154 (SageMath)
%o A178154 def h(p):
%o A178154 if not (is_prime(p) and mod(p, 4) == 3): return 0
%o A178154 return sum(kronecker(j, p) for j in range(1, (p + 1)//2))
%o A178154 aList = lambda upto: [h(n) for n in range(2, upto) if h(n) > 0]
%o A178154 print(aList(944)) # _Peter Luschny_, Jun 07 2025
%Y A178154 Cf. A178153, A002145.
%K A178154 nonn
%O A178154 1,3
%A A178154 _T. D. Noe_, May 21 2010
%E A178154 Name simplified by _Peter Luschny_, Jun 07 2025