A048153 a(n) = Sum_{k=1..n} (k^2 mod n).

0, 1, 2, 2, 10, 13, 14, 12, 24, 45, 44, 38, 78, 77, 70, 56, 136, 129, 152, 130, 182, 209, 184, 148, 250, 325, 288, 294, 406, 365, 372, 304, 484, 561, 490, 402, 666, 665, 572, 540, 820, 805, 860, 726, 840, 897, 846, 680, 980, 1125, 1156, 1170, 1378, 1305, 1210

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

See A048152 for the array T[n,k] = k^2 mod n.

Starting with a(2)=1 each 4th term is odd: a(n=2+4*k) = 1, 13, 45, 77, 129, 209, 325, 365, ... - Zak Seidov, Apr 22 2009

Positions of squares in A048153: 1, 2, 33, 51, 69, 105, 195, 250, 294, 1250, 4913, 9583, 13778, 48778, 65603, 83521.

Corresponding values of squares are: {0, 1, 22, 34, 46, 70, 130, 175, 203, 875, 3468, 6734, 9711, 34481, 46308, 58956}^2 = {0, 1, 484, 1156, 2116, 4900, 16900, 30625, 41209, 765625, 12027024, 45346756, 94303521, 1188939361, 2144430864, 3475809936}. - Zak Seidov, Nov 02 2011

For n > 1 also row sums of A060036. - Reinhard Zumkeller, Apr 29 2013

Conjecture: a(n) <= (n^2-1)/2. - Aspen A.M. Meissner, Mar 06 2025

Examples

			a(5) = 1^2 + 2^2 + (3^2 mod 5) + (4^2 mod 5) + (5^2 mod 5) = 1 + 4 + 4 + 1 + 0 = 10. (It is easily seen that the last term, n^2 mod n, is always zero and would not need to be included.) - _M. F. Hasler_, Oct 21 2013
		

Links

• Zak Seidov, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000330, A048152, A215573.

Programs

Haskell

a048153 = sum . a048152_row  -- Reinhard Zumkeller, Apr 29 2013

Mathematica

Table[Sum[PowerMod[k,2,n], {k,n-1}], {n,1,10000}] (* Zak Seidov, Nov 02 2011 *)

PARI

a(n)=sum(k=1,n,k^2%n) \\ Charles R Greathouse IV, Oct 21 2013

Python

def A048153(n): return sum(k**2%n for k in range(1,n)) # Chai Wah Wu, Jun 02 2024

Formula

a(n) == n*(n+1)*(2n+1)/6 (mod n). - Charles R Greathouse IV, Dec 28 2011

a(n) == n*(n-1)*(2n-1)/6 (mod n). - Chai Wah Wu, Jun 02 2024

a(n) mod n = A215573(n). - Alois P. Heinz, Jun 03 2024

Extensions

Definition made more explicit by M. F. Hasler, Oct 21 2013