A178154 -id:A178154 - cp's OEIS frontend

A178153 Difference between the numbers of quadratic residues (mod p) less than p/2 and greater than p/2, where p=prime(n).

1, 0, 1, 3, 0, 0, 3, 3, 0, 3, 0, 0, 3, 5, 0, 9, 0, 3, 7, 0, 5, 9, 0, 0, 0, 5, 9, 0, 0, 5, 15, 0, 9, 0, 7, 0, 3, 11, 0, 15, 0, 13, 0, 0, 9, 9, 7, 15, 0, 0, 15, 0, 21, 0, 13, 0, 11, 0, 0, 9, 0, 9, 19, 0, 0, 9, 0, 15, 0, 0, 19, 9, 0, 9, 17, 0, 0, 0, 0, 27, 0, 21, 0, 15, 15, 0, 0, 0, 7, 21, 25, 7, 27, 9, 21, 0

Offset: 2

T. D. Noe, May 21 2010

When prime(n)=1 (mod 4), then a(n)=0. When prime(n)=3 (mod 4), then a(n)>0. When prime(n)=3 (mod 8) and prime(n)>3, then 3 divides a(n). See Borevich and Shafarevich. The nonzero terms of this sequence are closely related to A002143, the class number of primes p=3 (mod 4).

Same as difference between the numbers of quadratic residues and nonresidues (mod p) less than p/2, where p=prime(n). - Jonathan Sondow, Oct 30 2011

			The quadratic residues of 19, the 8th prime, are 1, 4, 5, 6, 7, 9, 11, 16, 17. Hence a(8)=6-3=3.

Z. I. Borevich and I. R. Shafarevich, Number Theory, Academic Press, NY, 1966, p. 346.
H. Davenport, Multiplicative Number Theory, Graduate Texts in Math. 74, 2nd ed., Springer, 1980, p. 51.

R. J. Mathar, Table of n, a(n) for n = 2..2066
MathOverflow, Most squares in the first half-interval

Cf. A002143, A178154 (without the zero terms).

A178153 := 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:
    for r from ceil(p/2) to p-1 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&]] - Length[Select[Range[(p+1)/2,p-1], JacobiSymbol[ #,p]==1&]], {n,2,100}]
Table[p = Prime[n]; Sum[ JacobiSymbol[a, p], {a, 1, (p-1)/2}], {n, 2, 100}] (* Jonathan Sondow, Oct 30 2011 *)

a(n) = A178151(n) - A178152(n).

a(n) = Sum_{j=1..(p-1)/2} (j|p), where p = prime(n) and (j|p) = +-1 is the Legendre symbol. - Jonathan Sondow, Oct 30 2011

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}]

A186708 Number of quadratic residues (mod p) in the interval [1,2k+1], for primes p=4k+3.

Original entry on oeis.org

1, 2, 4, 6, 7, 9, 12, 14, 19, 18, 21, 22, 25, 28, 31, 34, 40, 39, 41, 42, 47, 52, 54, 54, 57, 59, 64, 67, 73, 72, 73, 75, 81, 87, 87, 94, 99, 96, 99, 104, 118, 118, 117, 118, 119, 127, 132, 125, 136, 129, 136, 138, 141, 154, 150, 157, 162

Offset: 1

M. F. Hasler, Feb 25 2011

nonn

For primes of the form p=4k+3 (A002145), count numbers in [1,2k+1] which are quadratic residues mod p.

R. K. Guy asks whether there is an elementary proof for the fact that there are always less quadratic residues in the interval [2k+2,4k+2] than in [1,2k+1].

Cf. A002145, A104635, A186709, A178154.

forprime( p=1,499, p%4==3|next; u=3; c=[1,0]; for(i=2,p-2, bittest(u,i^2%p) & next; u+=1<<(i^2%p); c[i^2%p*2\p+1]++); print1(c[1]", "))

a(n) = A104635(n) - A186709(n) = A186709(n) + A178154(n) = (A104635(n) + A178154(n))/2 = (A002145(n) + 2*A178154(n) - 1)/4.

A186709 Number of quadratic residues (mod p) in the interval [2k+2,4k+2], for primes p=4k+3.

Original entry on oeis.org

0, 1, 1, 3, 4, 6, 9, 9, 10, 15, 14, 17, 16, 23, 22, 29, 25, 30, 34, 39, 36, 37, 41, 45, 48, 52, 49, 52, 52, 59, 62, 66, 72, 68, 78, 79, 80, 87, 90, 87

Offset: 1

M. F. Hasler, Feb 25 2011

nonn

Cf. A002145, A186708, A178154.

forprime( p=1, 399, p%4==3|next; u=3; c=[1, 0]; for(i=2, p-2, bittest(u, i^2%p) & next; u+=1<<(i^2%p); c[i^2%p*2\p+1]++); print1(c[2]", "))

a(n) = A104635(n) - A186708(n) = A186708(n) - A178154(n) = (A104635(n) - A178154(n))/2 = (A002145(n) - 2*A178154(n) -1)/4.

cp's OEIS Frontend

A178153 Difference between the numbers of quadratic residues (mod p) less than p/2 and greater than p/2, where p=prime(n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

A186708 Number of quadratic residues (mod p) in the interval [1,2k+1], for primes p=4k+3.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula

A186709 Number of quadratic residues (mod p) in the interval [2k+2,4k+2], for primes p=4k+3.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula