A048153 -id:A048153 - cp's OEIS frontend

A199117 Integer values of A048153(n)/n in the order they appear in A048153.

0, 2, 2, 4, 6, 8, 8, 8, 10, 14, 12, 14, 18, 20, 20, 18, 20, 26, 22, 26, 30, 32, 32, 28, 36, 36, 34, 38, 42, 44, 42, 38, 48, 50, 46, 50, 54, 56, 52, 48, 54, 60, 58, 60, 64, 68, 66, 60, 72, 74, 68, 70, 78, 76, 80, 72, 78, 86, 76, 84, 90, 92

Offset: 1

Zak Seidov, Nov 03 2011

nonn

Not all even integers are here, first absent values are 16, 24, 40, 62, 88, 94, 100, 106, 112, 130, 142, 144, 154, 164, 172, 180.

Zak Seidov, Table of n, a(n) for n = 1..1001 [Offset shifted by Georg Fischer, Sep 01 2021]

Cf. A007310, A048153.

a(n) = A048153(m)/m, m = A007310(n).

Offset corrected by Georg Fischer, Sep 01 2021

A199330 Squares in A048153.

Original entry on oeis.org

0, 1, 484, 1156, 2116, 4900, 16900, 30625, 41209, 765625, 12027024, 45346756, 94303521, 1188939361, 2144430864, 3475809936, 6168531600, 26159180644, 211618400400, 483560298225, 999898002601, 1722945012100, 3559421809449, 15912878919025, 19806300482329, 50911350177961

Offset: 1

Zak Seidov, Nov 05 2011

nonn

Corresponding values of square roots are 0, 1, 22, 34, 46, 70, 130, 175, 203, 875, 3468, 6734, 9711, 34481, 46308, 58956, 78540, 161738, 460020, 695385, 999949, 1312610, 1886643, 3989095, 4450427, 7135219.

Intersection of A000290 and A048153.

Cf. A199551.

a(n) = A048153(A199551(n)). - Amiram Eldar, Apr 08 2025

a(17)-a(20) from Charles R Greathouse IV, Nov 04 2011

Offset corrected and a(21)-a(26) added by Amiram Eldar, Apr 08 2025

A199551 Positions of squares in A048153.

Original entry on oeis.org

1, 2, 33, 51, 69, 105, 195, 250, 294, 1250, 4913, 9583, 13778, 48778, 65603, 83521, 111265, 228939, 651695, 984150, 1414562, 1857848, 2668626, 5643110, 6295726, 10091354

Offset: 1

Zak Seidov, Nov 08 2011

Cf. A048153, A199330.

A048153(a(n)) = A199330(n).

a(21)-a(26) from Amiram Eldar, Apr 08 2025

A060036 Triangular array T read by rows: T(n,k) = k^2 mod n, for k = 1,2,...,n-1, n = 2,3,...

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 4, 4, 1, 1, 4, 3, 4, 1, 1, 4, 2, 2, 4, 1, 1, 4, 1, 0, 1, 4, 1, 1, 4, 0, 7, 7, 0, 4, 1, 1, 4, 9, 6, 5, 6, 9, 4, 1, 1, 4, 9, 5, 3, 3, 5, 9, 4, 1, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1, 1, 4, 9, 3, 12, 10, 10, 12, 3, 9, 4, 1, 1, 4, 9, 2, 11, 8, 7, 8, 11, 2, 9, 4, 1, 1, 4, 9, 1, 10, 6, 4, 4, 6, 10

Offset: 2

N. J. A. Sloane, Mar 17 2001

T(n,k) = A048152(n-1,k), 1 <= k < n; T(2*n-1,n-1) = A123684(n-1) = A225126(n-1). - Reinhard Zumkeller, Apr 29 2013

			The triangle T(n,k) begins:
n\k 1 2 3 4 5 6 7 8 9 10 11 ...
-------------------------------
2:  1
3:  1 1
4:  1 0 1
5:  1 4 4 1
6:  1 4 3 4 1
7:  1 4 2 2 4 1
6:  1 4 1 0 1 4 1
9:  1 4 0 7 7 0 4 1
10: 1 4 9 6 5 6 9 4 1
11: 1 4 9 5 3 3 5 9 4  1
12: 1 4 9 4 1 0 1 4 9  4  1
...  reformatted by - _Wolfdieter Lang_, Dec 17 2018

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

Cf. A048152, A060037.

Cf. A048153 (row sums).

a060036 n k = a060036_tabl !! (n-2) !! (k-1)
a060036_row n = a060036_tabl !! (n-2)
a060036_tabl = map init $ tail a048152_tabl
-- Reinhard Zumkeller, Apr 29 2013

Flatten[Table[PowerMod[k,2,n],{n,2,20},{k,n-1}]] (* Harvey P. Dale, Feb 27 2012 *)

{ n=1; for (m=2, 46, for (k=1, m-1, write("b060036.txt", n++, " ", k^2 % m)); ) } \\ Harry J. Smith, Jul 01 2009

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

A215573 a(n) = n(n+1)(2n+1)/6 modulo n.

Original entry on oeis.org

0, 1, 2, 2, 0, 1, 0, 4, 6, 5, 0, 2, 0, 7, 10, 8, 0, 3, 0, 10, 14, 11, 0, 4, 0, 13, 18, 14, 0, 5, 0, 16, 22, 17, 0, 6, 0, 19, 26, 20, 0, 7, 0, 22, 30, 23, 0, 8, 0, 25, 34, 26, 0, 9, 0, 28, 38, 29, 0, 10, 0, 31, 42, 32, 0, 11, 0, 34, 46, 35, 0, 12, 0, 37, 50, 38

Offset: 1

Zak Seidov, Aug 16 2012

a(n) = 0 for n = 6k +- 1, that is, A007310 (numbers congruent to 1 or 5 mod 6).

Graph consists of 4 linear patterns.

Zak Seidov, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2,0,0,0,0,0,-1).

Cf. A000330, A007310, A048153.

seq(modp(n*(n+1)*(2*n+1)/6,n),n=1..100); # Muniru A Asiru, Feb 07 2019

Table[Mod[(n(n+1)(2n+1))/6,n],{n,80}] (* or *) LinearRecurrence[{0,0,0,0,0,2,0,0,0,0,0,-1},{0,1,2,2,0,1,0,4,6,5,0,2},80] (* Harvey P. Dale, Aug 25 2023 *)

a(n)=n*(n+1)*(2*n+1)/6 % n; \\ Michel Marcus, Oct 19 2013

concat(0, Vec(x^2*(1 + 2*x + 2*x^2 + x^4 + 2*x^6 + 2*x^7 + x^8) / ((1 - x)^2*(1 + x)^2*(1 - x + x^2)^2*(1 + x + x^2)^2) + O(x^80))) \\ Colin Barker, Feb 07 2019

def A215573(n): return n*(n-1)*((n<<1)-1)//6%n # Chai Wah Wu, Jun 03 2024

a(n) = A000330(n) mod n.
From Colin Barker, Feb 07 2019: (Start)
G.f.: x^2*(1 + 2*x + 2*x^2 + x^4 + 2*x^6 + 2*x^7 + x^8) / ((1 - x)^2*(1 + x)^2*(1 - x + x^2)^2*(1 + x + x^2)^2).
a(n) = 2*a(n-6) - a(n-12) for n>12. (End)
a(6*n) = n, a(6*n+1) = 0, a(6*n+2) = 3*n+1, a(6*n+3) = 4*n+2, a(6*n+4) = 3*n+2, a(6*n+5) = 0. - Philippe Deléham, Mar 05 2023
a(n) = A048153(n) mod n. - Alois P. Heinz, Jun 03 2024
a(n) = A000330(n-1) mod n. - Chai Wah Wu, Jun 03 2024
Sum_{k=1..n} a(k) ~ (11/72) * n^2. - Amiram Eldar, Apr 05 2025

A373749 Triangle read by rows: T(n, k) = MOD(k^2, n) where MOD(a, n) = a if n = 0 and otherwise a - n*floor(a/n). (Quadratic residue.)

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Jun 23 2024

The definition of the binary operation MOD in the name follows CMath (Graham et al.) and Bach & Shallit. This is important because some CAS unfortunately do not follow this definition and throw a 'division by zero' error if n = 0.

Row n reduced to a set is the set of the quadratic residues mod n.

			Triangle starts:
  [0] 0;
  [1] 0, 0;
  [2] 0, 1, 0;
  [3] 0, 1, 1, 0;
  [4] 0, 1, 0, 1, 0;
  [5] 0, 1, 4, 4, 1, 0;
  [6] 0, 1, 4, 3, 4, 1, 0;
  [7] 0, 1, 4, 2, 2, 4, 1, 0;
  [8] 0, 1, 4, 1, 0, 1, 4, 1, 0;
  [9] 0, 1, 4, 0, 7, 7, 0, 4, 1, 0;
 [10] 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0;

Eric Bach and Jeffrey Shallit, Algorithmic Number Theory, p. 21.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., pp. 81f.

Variants: A048152, A096008.

Cf. A048153 (row sums), A373750 (middle terms).

Mod(n, k) = k == 0 ? n : mod(n, k)
T(n, k) = Mod(k^2, n)
for n in 0:10
    [T(n, k) for k in 0:n] |> println
end

REM := (n, k) -> ifelse(k = 0, n, irem(n, k)):
T := n -> local k; seq(REM(k^2, n), k = 0..n):
seq(T(n), n = 0..12);

MOD[n_, k_] := If[k == 0, n, Mod[n, k]];
Table[MOD[k^2, n], {n, 0, 10}, {k, 0, n}]

def A373749(n, k): return mod(k^2, n)
for n in range(11): print([A373749(n, k) for k in range(n + 1)])

A230366 a(n) = Sum_{k=1..floor(n/2)} (k^2 mod n).

Original entry on oeis.org

0, 1, 1, 1, 5, 8, 7, 6, 12, 25, 22, 19, 39, 42, 35, 28, 68, 69, 76, 65, 91, 110, 92, 74, 125, 169, 144, 147, 203, 190, 186, 152, 242, 289, 245, 201, 333, 342, 286, 270, 410, 413, 430, 363, 420, 460, 423, 340, 490, 575, 578, 585, 689, 666, 605, 546, 760, 841

Offset: 1

Jon Perry, Oct 17 2013

nonn

a(26) and a(27) are both squares. Conjecture: the number of n such that a(n) and a(n+1) are both squares is infinite.

a(p = prime) == 0 (mod p) for p > 3.

Cf. A048153.

for (i=1;i<50;i++) {
c=0;
for (j=1;j<=i/2;j++) c+=(j*j)%i;
document.write(c+", ");
}

Table[Sum[Mod[k^2, n], {k, Floor[n/2]}], {n, 100}] (* T. D. Noe, Oct 22 2013 *)
Table[Sum[PowerMod[k,2,n],{k,Floor[n/2]}],{n,100}] (* Harvey P. Dale, Jul 03 2022 *)

a(n)=sum(i=1,floor(n/2),(i*i)%n) \\ Ralf Stephan, Oct 19 2013

def A230366(n): return sum(k**2%n for k in range(1,(n>>1)+1)) # Chai Wah Wu, Jun 02 2024

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);
```

A352620 Irregular triangle read by rows which are rows of successive n X n matrices M(n) with entries M(n)[i,j] = i*j mod n+1.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 2, 3, 2, 0, 2, 3, 2, 1, 1, 2, 3, 4, 2, 4, 1, 3, 3, 1, 4, 2, 4, 3, 2, 1, 1, 2, 3, 4, 5, 2, 4, 0, 2, 4, 3, 0, 3, 0, 3, 4, 2, 0, 4, 2, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 2, 4, 6, 1, 3, 5, 3, 6, 2, 5, 1, 4, 4, 1, 5, 2, 6, 3, 5, 3, 1, 6, 4, 2, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5

Offset: 1

Luca Onnis, Mar 24 2022

Each matrix represents all possible products between the elements of Z_(n+1), where Z_k is the ring of integers mod k.
Those matrices are symmetric.
The first row is equal to the first column which is equal to 1,2,...,n.

			Matrices begin:
  n=1:  1,
  n=2:  1, 2,
        2, 1,
  n=3:  1, 2, 3,
        2, 0, 2,
        3, 2, 1,
  n=4:  1, 2, 3, 4,
        2, 4, 1, 3,
        3, 1, 4, 2,
        4, 3, 2, 1;
For example, the 6 X 6 matrix generated by Z_7 is the following:
  1 2 3 4 5 6
  2 4 6 1 3 5
  3 6 2 5 1 4
  4 1 5 2 6 3
  5 3 1 6 4 2
  6 5 4 3 2 1
The trace of this matrix is 14 = A048153(7).

Onno Cain, Gaussian Integers, Rings, Finite Fields, and the Magic Square of Squares, arXiv:1908.03236 [math.RA], 2019.
Matt Parker and Brady Haran, Finite Fields & Return of The Parker Square, Numberphile video (Oct 7, 2021).

Cf. A048153 (traces), A349099 (permanents), A160255 (sum entries), A088922 (ranks).

Cf. A074930.

Flatten[Table[Table[Mod[k*Table[i, {i, 1, p - 1}], p], {k, 1, p - 1}], {p, 1, 10}]]

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

A199117 Integer values of A048153(n)/n in the order they appear in A048153.

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A199330 Squares in A048153.

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A199551 Positions of squares in A048153.

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A060036 Triangular array T read by rows: T(n,k) = k^2 mod n, for k = 1,2,...,n-1, n = 2,3,...

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A215573 a(n) = n*(n+1)*(2n+1)/6 modulo n.

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A373749 Triangle read by rows: T(n, k) = MOD(k^2, n) where MOD(a, n) = a if n = 0 and otherwise a - n*floor(a/n). (Quadratic residue.)

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A230366 a(n) = Sum_{k=1..floor(n/2)} (k^2 mod n).

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

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A215573 a(n) = n(n+1)(2n+1)/6 modulo n.