A178154 Sum_{a=1..(p-1)/2} LegendreSymbol(a, p) for primes p == 3 (mod 4).

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, 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, 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

Offset: 1

Author

T. D. Noe, May 21 2010

Keywords

• nonn

Comments

Wenpeng Zhang proves that a(n) > 0. - Peter Luschny, Jun 07 2025

References

• Richard Kenneth Guy, Unsolved Problems in Number Theory, Springer-Verlag, Berlin, 1994, pp. 244.

Links

• Wenpeng Zhang, An old number theory problem related to the Legendre symbol, arXiv:2506.02434 [math.HO], Jun. 2025.

Crossrefs

Cf. A178153, A002145.

Programs

Maple

P := upto -> select(n -> isprime(n) and modp(n, 4) = 3, [seq(1..upto)]):
a := p -> add(NumberTheory:-LegendreSymbol(a, p), a = 1..(p-1)/2):
seq(a(p), p in P(940));  # Peter Luschny, Jun 07 2025

Mathematica

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&]}]

SageMath

def h(p):
    if not (is_prime(p) and mod(p, 4) == 3): return 0
    return sum(kronecker(j, p) for j in range(1, (p + 1)//2))
aList = lambda upto: [h(n) for n in range(2, upto) if h(n) > 0]
print(aList(944))  # Peter Luschny, Jun 07 2025

Formula

A178153 for primes p=3 (mod 4).

Extensions

Name simplified by Peter Luschny, Jun 07 2025