A178153 -id:A178153 - cp's OEIS frontend

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

T. D. Noe, May 21 2010

nonn

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

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

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

Cf. A178153, A002145.

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

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&]}]

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

A178153 for primes p=3 (mod 4).

Name simplified by Peter Luschny, Jun 07 2025

A178151 The number of quadratic residues (mod p) less than p/2, where p=prime(n).

Original entry on oeis.org

1, 1, 2, 4, 3, 4, 6, 7, 7, 9, 9, 10, 12, 14, 13, 19, 15, 18, 21, 18, 22, 25, 22, 24, 25, 28, 31, 27, 28, 34, 40, 34, 39, 37, 41, 39, 42, 47, 43, 52, 45, 54, 48, 49, 54, 57, 59, 64, 57, 58, 67, 60, 73, 64, 72, 67, 73, 69, 70, 75, 73, 81, 87, 78, 79, 87, 84, 94, 87, 88, 99, 96, 93

Offset: 2

T. D. Noe, May 21 2010

nonn

Sequence A063987 lists the quadratic residues (mod p) for each prime p. When p=1 (mod 4), there are an equal number of quadratic residues less than p/2 and greater than p/2. When p=3 (mod 4), there are always more quadratic residues less than p/2 than greater than p/2.

			The quadratic residues of 19, the 8th prime, are 1, 4, 5, 6, 7, 9, 11, 16, 17. Six of these are less than 19/2. Hence a(8)=6.

R. J. Mathar, Table of n, a(n) for n = 2..3132
MathOverflow, Most squares in the first half-interval

Cf. A178152, A178153, A178154

A178151 := proc(n)
    local r,a,p;
    p := ithprime(n) ;
    a := 0 ;
    for r from 1 to p/2 do
        if numtheory[legendre](r,p) =1 then
            a := a+1 ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, Feb 10 2017

Table[p=Prime[n]; Length[Select[Range[(p-1)/2], JacobiSymbol[ #,p]==1&]], {n,2,100}]

A178152 The number of quadratic residues (mod p) greater than p/2, where p=prime(n).

Original entry on oeis.org

0, 1, 1, 1, 3, 4, 3, 4, 7, 6, 9, 10, 9, 9, 13, 10, 15, 15, 14, 18, 17, 16, 22, 24, 25, 23, 22, 27, 28, 29, 25, 34, 30, 37, 34, 39, 39, 36, 43, 37, 45, 41, 48, 49, 45, 48, 52, 49, 57, 58, 52, 60, 52, 64, 59, 67, 62, 69, 70, 66, 73, 72, 68, 78, 79, 78, 84, 79, 87, 88, 80, 87, 93, 90

Offset: 2

T. D. Noe, May 21 2010

nonn

Sequence A063987 lists the quadratic residues (mod p) for each prime p. When p=1 (mod 4), there are an equal number of quadratic residues less than p/2 and greater than p/2. When p=3 (mod 4), there are always more quadratic residues less than p/2 than greater than p/2.

			The quadratic residues of 19, the 8th prime, are 1, 4, 5, 6, 7, 9, 11, 16, 17. Three of these are greater than 19/2. Hence a(8)=3.

MathOverflow, Most squares in the first half-interval

Cf. A178151, A178153, A178154

Table[p=Prime[n]; Length[Select[Range[(p+1)/2,p-1], JacobiSymbol[ #,p]==1&]], {n,2,100}]

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

SageMath

Formula

Extensions

A178151 The number of quadratic residues (mod p) less than p/2, where p=prime(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A178152 The number of quadratic residues (mod p) greater than p/2, where p=prime(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica