A048152 -id:A048152 - cp's OEIS frontend

A060037 Triangular array T read by rows: T(n,k)=k^2 mod n, for k=1,2,...,[n/2], n=2,3,...

1, 1, 1, 0, 1, 4, 1, 4, 3, 1, 4, 2, 1, 4, 1, 0, 1, 4, 0, 7, 1, 4, 9, 6, 5, 1, 4, 9, 5, 3, 1, 4, 9, 4, 1, 0, 1, 4, 9, 3, 12, 10, 1, 4, 9, 2, 11, 8, 7, 1, 4, 9, 1, 10, 6, 4, 1, 4, 9, 0, 9, 4, 1, 0, 1, 4, 9, 16, 8, 2, 15, 13, 1, 4, 9, 16, 7, 0, 13, 10, 9, 1, 4, 9, 16, 6, 17, 11, 7, 5, 1, 4, 9, 16, 5, 16

Offset: 2

N. J. A. Sloane, Mar 17 2001

			1; 1; 1,0; 1,4; 1,4,3; 1,4,2; ...

T. D. Noe, Rows n=2..100 of triangle, flattened

Cf. A048152, A060036.

Flatten[Table[PowerMod[k,2,n],{n,2,20},{k,Floor[n/2]}]] (* Harvey P. Dale, Mar 05 2012 *)

More terms from Larry Reeves (larryr(AT)acm.org), Mar 20 2001

A231589 a(n) = sum_{k=1..(n-1)/2} (k^2 mod n).

Original entry on oeis.org

0, 0, 1, 1, 5, 5, 7, 6, 12, 20, 22, 19, 39, 35, 35, 28, 68, 60, 76, 65, 91, 99, 92, 74, 125, 156, 144, 147, 203, 175, 186, 152, 242, 272, 245, 201, 333, 323, 286, 270, 410, 392, 430, 363, 420, 437, 423, 340, 490, 550, 578, 585, 689, 639, 605, 546, 760, 812

Offset: 1

Michel Marcus, Nov 11 2013

nonn

This expression occurred to S. A. Shirali while demonstrating a result concerning A081115 and A228432. This led him to investigate integers n such that a(n) = n*(n-1)/4, a(n) = floor(n/4), or a(n) = n*(n-1)/4 - n.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
S. A. Shirali, A family portrait of primes-a case study in discrimination, Math. Mag. Vol. 70, No. 4 (Oct., 1997), pp. 263-272.

Cf. A048152, A048153.

Table[Sum[PowerMod[k,2,n],{k,(n-1)/2}],{n,60}] (* Harvey P. Dale, Jan 30 2016 *)

PARI
```
a(n) = sum(k=1, (n-1)\2, k^2 % n);
```

A306271 a(n) is the smallest positive integer x such that x^2 mod n is a square, with x^2 >= n.

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 5, 3, 3, 7, 8, 4, 10, 11, 4, 4, 13, 6, 15, 6, 5, 18, 19, 5, 5, 21, 6, 9, 24, 8, 26, 6, 7, 29, 6, 6, 31, 32, 8, 7, 35, 10, 37, 16, 7, 40, 41, 7, 7, 10, 10, 19, 46, 12, 8, 9, 14, 51, 52, 8, 54, 55, 8, 8, 9, 14, 59, 26, 16, 12, 63, 9, 65, 66, 10

Offset: 1

Daniel Suteu, Feb 01 2019

nonn

			For n = 10, a(10) = 7, which is the smallest positive integer x such that x^2 mod n is a square and that x^2 >= n. Here 7^2 mod 10 = 9 = 3^2.

Daniel Suteu, Table of n, a(n) for n = 1..10000
Wikipedia, Congruence of squares

Cf. A000290, A048152, A062803, A112045, A254328, A306257.

a:= proc(n) local k, t;
      for k do t:= irem(k^2, n);
        if issqr(t) and isqrt(t)<>k then break fi
      od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 01 2019

a[n_] := For[x = Sqrt[n]//Ceiling, True, x++, If[IntegerQ[Sqrt[PowerMod[x, 2, n]]], Return[x]]];
Array[a, 100] (* Jean-François Alcover, Nov 07 2020 *)

a(n) = for(k=sqrtint(n), oo, if(issquare(k^2 % n) && sqrtint(k^2 % n) != k, return(k)));

a(n^2) = n.

a(p) = p - floor(sqrt(p)), for prime p > 2.

A343720 Triangle read by rows: T(n,k) = k^2 mod n for k = 0..n-1, n >= 1.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 4, 4, 1, 0, 1, 4, 3, 4, 1, 0, 1, 4, 2, 2, 4, 1, 0, 1, 4, 1, 0, 1, 4, 1, 0, 1, 4, 0, 7, 7, 0, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 5, 3, 3, 5, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 3, 12, 10, 10, 12, 3, 9, 4, 1

Offset: 1

Jon E. Schoenfield, Apr 26 2021

Similar to A048152 and A060036, but each row in this sequence begins at k = 0 and ends at k = n-1 (the minimum and maximum residues modulo n, respectively).

			Triangle begins:
  n\k| 0  1  2  3  4  5  6  7  8  9 10 11
  ---+-----------------------------------
   1 | 0
   2 | 0, 1
   3 | 0, 1, 1
   4 | 0, 1, 0, 1
   5 | 0, 1, 4, 4, 1
   6 | 0, 1, 4, 3, 4, 1
   7 | 0, 1, 4, 2, 2, 4, 1
   8 | 0, 1, 4, 1, 0, 1, 4, 1
   9 | 0, 1, 4, 0, 7, 7, 0, 4, 1
  10 | 0, 1, 4, 9, 6, 5, 6, 9, 4, 1
  11 | 0, 1, 4, 9, 5, 3, 3, 5, 9, 4, 1
  12 | 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
Eric Weisstein's World of Mathematics, Quadratic Residue

Cf. A048152, A060036.

T(n,k) = k^2 % n \\ Andrew Howroyd, Jan 05 2024

T(n,k) = k^2 mod n.

T(n,k) = T(n,n-k).

A306284 a(n) is the smallest positive integer x such that x > y >= 0 and n divides x^2 - y^2.

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 4, 3, 3, 6, 6, 4, 7, 8, 4, 4, 9, 6, 10, 6, 5, 12, 12, 5, 5, 14, 6, 8, 15, 8, 16, 6, 7, 18, 6, 6, 19, 20, 8, 7, 21, 10, 22, 12, 7, 24, 24, 7, 7, 10, 10, 14, 27, 12, 8, 9, 11, 30, 30, 8, 31, 32, 8, 8, 9, 14, 34, 18, 13, 12, 36, 9, 37, 38, 10, 20, 9, 16, 40, 9, 9, 42, 42, 10, 11, 44, 16

Offset: 1

Jianing Song, Feb 03 2019

Different from A306271: here x^2 mod n is not necessarily a square. For most n, a(n) != A306271(n).
It seems that n divides a(n)^2 if and only if n divides A306271(n)^2.
a(n) >= sqrt(n) with equality if and only if n is a square. - Robert Israel, Feb 05 2019

			a(10) = 6 because 10 divides 6^2 - 4^2 = 10, and 6 is the smallest possible value for x such that x > y >= 0 and that 10 divides x^2 - y^2.
a(87) = 16 because 87 divides 16^2 - 13^2 = 87, and 16 is the smallest possible value for x such that x > y >= 0 and that 87 divides x^2 - y^2.

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

Cf. A048152, A306271.

f:= proc(n) local S, x,t;
S:= {0}:
for x from 1 do
  t:= x^2 mod n;
  if member(t,S) then return x
    else S:= S union {t}
  fi
od
end proc:
map(f, [$1..100]); # Robert Israel, Feb 05 2019

a(n) = for(x=1, n, for(y=0, x-1, if((x^2-y^2)%n==0, return(x))))

from itertools import count
def A306284(n):
    y, a = 0, set()
    for x in count(0):
        if y in a: return x
        a.add(y)
        y = (y+(x<<1)+1)%n # Chai Wah Wu, Apr 25 2024

a(n^2) = n.
a(p) = (p + 1)/2 for primes p > 2.
For odd primes p and q, a(p*q) = (p+q)/2. - Robert Israel, Feb 08 2019

A344851 a(n) = (n^2) mod (2^A070939(n)).

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 4, 1, 0, 1, 4, 9, 0, 9, 4, 1, 0, 1, 4, 9, 16, 25, 4, 17, 0, 17, 4, 25, 16, 9, 4, 1, 0, 1, 4, 9, 16, 25, 36, 49, 0, 17, 36, 57, 16, 41, 4, 33, 0, 33, 4, 41, 16, 57, 36, 17, 0, 49, 36, 25, 16, 9, 4, 1, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Offset: 0

Rémy Sigrist, May 30 2021

Informally, if n has w binary digits, a(n) is obtained by keeping the w final binary digits of n^2.
For n > 0, a(n) is the final digit of n^2 in base A062383(n).
This sequence has interesting graphical features (see illustration in Links section).

			For n = 42:
- A070939(42) = 6,
- a(42) = (42^2) mod (2^6) = 1764 mod 64 = 36.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
John S. McCaskill and Peter R.Wills, On permutations derived from integer powers x^n, arXiv:1907.01890 [math.NT], 2019.
Rémy Sigrist, Scatterplot of the sequence for n = 2^18..2^19

Cf. A000290, A048152, A062383, A070939, A086341, A116882, A316347 (decimal analog).

{0}~Join~Table[Mod[n^2,2^(1+Floor@Log2@n)],{n,100}] (* Giorgos Kalogeropoulos, Jun 02 2021 *)

PARI
```
a(n) = (n^2) % 2^#binary(n)
```

def a(n): return (n**2) % (2**n.bit_length())
print([a(n) for n in range(75)]) # Michael S. Branicky, May 30 2021

a(n) = 0 iff n = 0 or n > 1 and n belongs to A116882.
a(n) = 1 iff n belongs to A086341.
a(2^k + m) = a(2^(k+1)-m) for any k > 0 and m = 0..2^k.

A228587 Sum of the squares (modulo n) of the odd numbers less than n.

Original entry on oeis.org

0, 1, 1, 2, 5, 5, 7, 4, 12, 25, 22, 22, 39, 49, 35, 40, 68, 69, 76, 50, 91, 77, 92, 44, 125, 169, 144, 182, 203, 205, 186, 208, 242, 289, 245, 210, 333, 285, 286, 180, 410, 413, 430, 374, 420, 529, 423, 376, 490, 625, 578, 546, 689, 585, 605, 476, 760, 841, 767, 710

Offset: 1

R. J. Mathar, Aug 27 2013

Sum over the odd-numbered columns in row n of A048152.

Cf. A000447.

A228587 := proc(n)
        local a,o ;
        a := 0 ;
        for o from 1 to n-1 by 2 do
                a := a+ modp(o^2,n) ;
        end do:
        a ;
end proc:

a(n) = sum_{k=1,3,5,...,n-1} (k^2 mod n).
a(n) <= A048153(n).
A187468(n) = a(2^n).

cp's OEIS Frontend

A060037 Triangular array T read by rows: T(n,k)=k^2 mod n, for k=1,2,...,[n/2], n=2,3,...

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A231589 a(n) = sum_{k=1..(n-1)/2} (k^2 mod n).

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A306271 a(n) is the smallest positive integer x such that x^2 mod n is a square, with x^2 >= n.

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A343720 Triangle read by rows: T(n,k) = k^2 mod n for k = 0..n-1, n >= 1.

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A306284 a(n) is the smallest positive integer x such that x > y >= 0 and n divides x^2 - y^2.

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A344851 a(n) = (n^2) mod (2^A070939(n)).

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A228587 Sum of the squares (modulo n) of the odd numbers less than n.

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