A228131 -id:A228131 - cp's OEIS frontend

A014817 a(n) = Sum_{k=1..n} floor(k^2/n).

1, 2, 4, 7, 9, 13, 18, 24, 29, 34, 42, 51, 57, 67, 78, 90, 97, 110, 122, 137, 149, 163, 180, 198, 211, 226, 246, 265, 281, 303, 324, 348, 365, 386, 412, 439, 457, 483, 512, 540, 561, 590, 618, 651, 679, 709, 742

Offset: 1

N. J. A. Sloane

			Row sums of the underlying triangle of floor(k^2/n), 1<=k<=n:
1;
0,2;
0,1,3;
0,1,2,4;
0,0,1,3,5;
0,0,1,2,4,6;
0,0,1,2,3,5,7;
0,0,1,2,3,4,6,8;
0,0,1,1,2,4,5,7,9;
0,0,0,1,2,3,4,6,8,10;
- _R. J. Mathar_, Aug 09 2013

M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkhauser, 1985, p. 103.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A177041, A166387, A166375, A165993, A227841, A227842.

[(&+[Floor(k^2/n): k in [1..n]]): n in [1..50]]; // G. C. Greubel, May 10 2018

A014817 := m->sum( floor(k^2/m), k=1..m);

Table[Sum[Floor[k^2/n],{k,n}],{n,50}] (* Harvey P. Dale, Feb 23 2015 *)

A014817(n)=sum(k=1,n,k^2\n)  \\ M. F. Hasler, Dec 11 2010

a(n)=n^2-sum(m=1,n,sqrtint(n*m-1)) \\ Charles R Greathouse IV, Jun 20 2013

a(n) = n +A166375(n).

For prime p>2, a(p) = (p^2+2)/3 - A228131(p)/p. In particular, for prime p==1 (mod 4), a(p) = (p^2+2)/3. - Max Alekseyev, Aug 11 2013

A255643 Difference between sums of quadratic residues and non-residues modulo n that are coprime to n.

Original entry on oeis.org

0, 1, -1, -2, 0, -4, -7, -14, -3, 0, -11, -22, 0, 0, -50, -44, 0, -12, -19, -60, -84, -44, -69, -94, 0, 0, -9, -98, 0, -80, -93, -152, -176, 0, -280, -138, 0, -76, -312, -300, 0, -126, -43, -286, -330, 0, -235, -332, -49, 0, -476, -364, 0, -36, -660, -602, -570, 0, -177, -380, 0, 0, -630, -560, -780, -374, -67, -680, -782, -560, -497, -714, 0, 0, -850, -798, -1232, -468, -395, -1080, -27, 0, -249, -882, -1360, -172, -1508, -1430, 0, -600, -1820, -1058, -1674, 0, -2090, -1240, 0, 0, -1518, -1100

Offset: 1

Max Alekseyev, Mar 01 2015

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A228131, A255644.

f:= proc(n) local t;
  add(`if`(igcd(t,n)=1, t*numtheory:-quadres(t,n),0), t=1..n-1)
end proc:
map(f, [$1..100]); # Robert Israel, Feb 27 2025

{ A255643(n) = my(r); r=0; for(i=0,n-1, if(gcd(i,n)>1,next); if(issquare(Mod(i,n)), r+=i, r-=i) ); r }

For prime n, a(n) = A228131(n) = A255644(n).
For prime n==1 (mod 4), a(n) = 0.
For prime n==3 (mod 4) and n > 3, i.e., n=A002145(m) for m > 1, a(n) = -n*A002143(m).
Is 2 the only n for which a(n) > 0? - Robert Israel, Feb 27 2025

A255644 Difference between sums of quadratic residues and non-residues modulo n (residues are not necessarily coprime to n).

Original entry on oeis.org

0, 1, -1, -4, 0, 1, -7, -18, -12, 5, -11, -38, 0, -7, -45, -92, 0, -33, -19, -120, -70, -11, -69, -192, -50, 13, -99, -210, 0, -135, -93, -352, -198, 17, -245, -438, 0, -19, -325, -510, 0, -245, -43, -550, -540, -115, -235, -880, -196, -175, -459, -728, 0, -333, -715, -1036, -532, 29, -177, -1230, 0, -155, -1155, -1440, -780, -693, -67, -1292, -966, -875, -497, -1908, 0, 37, -1325, -1558, -1232, -1079, -395, -2500, -864, 41, -249, -2366, -1360, -43, -1479, -2552, 0, -2025, -1729, -2346, -1426, -423, -2185, -3568, 0, -637, -2673, -3000

Offset: 1

Max Alekseyev, Mar 01 2015

sign

Cf. A228131, A255643.

{ A255643(n) = my(r); r=0; for(i=0,n-1, if(issquare(Mod(i,n)), r+=i, r-=i) ); r }

For prime n, a(n) = A228131(n) = A255643(n).
For prime n==1 (mod 4), a(n) = 0.
For prime n==3 (mod 4) and n > 3, i.e., n=A002145(m) for m > 1, a(n) = -n*A002143(m).

A232597 a(n) = sum of odd k in 1..n for which Kronecker(k,n)=1.

Original entry on oeis.org

1, 1, 1, 4, 1, 6, 1, 8, 13, 13, 18, 8, 13, 31, 1, 64, 38, 25, 50, 40, 23, 78, 26, 24, 124, 131, 65, 84, 83, 94, 66, 128, 103, 158, 143, 216, 137, 212, 42, 160, 224, 156, 242, 176, 132, 327, 147, 176, 513, 297, 259, 312, 303, 246, 170, 224, 303, 365, 509, 240

Offset: 1

Antti Karttunen, Nov 26 2013

nonn

Restricted to primes: A232505(n) = a(A000040(n)). Cf. also A228131.

A232597 := proc(n)
    local a;
    a := 0 ;
    for k from 1 to n by 2 do
        if numtheory[jacobi](k,n) = 1 then
            a := a+k ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, May 25 2017

A232597(n) = {s=0; for(k=1, n, s=s+((k%2)*((1+kronecker(k, n))\2)*k)); return(s); }
for(n=1, 60, print1(A232597(n), ", "))

a(n)=my(s); forstep(k=1,n,if(kronecker(k,n)==1, s+=k)); s \\ Charles R Greathouse IV, Nov 26 2013

cp's OEIS Frontend

A014817 a(n) = Sum_{k=1..n} floor(k^2/n).

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A255643 Difference between sums of quadratic residues and non-residues modulo n that are coprime to n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

PARI

Formula

A255644 Difference between sums of quadratic residues and non-residues modulo n (residues are not necessarily coprime to n).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A232597 a(n) = sum of odd k in 1..n for which Kronecker(k,n)=1.

Views

Author

Keywords

Crossrefs

Programs

Maple

PARI

PARI