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URL: https://oeis.org/A048153

⇱ A048153 - OEIS

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A048153
a(n) = Sum_{k=1..n} (k^2 mod n).
15
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
1,3
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
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
EXAMPLE
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
MATHEMATICA
Table[Sum[PowerMod[k, 2, n], {k, n-1}], {n, 1, 10000}] (* Zak Seidov, Nov 02 2011 *)
PROG
(Haskell)
a048153 = sum . a048152_row -- Reinhard Zumkeller, Apr 29 2013
(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
CROSSREFS
KEYWORD
nonn
EXTENSIONS
Definition made more explicit by M. F. Hasler, Oct 21 2013
STATUS
approved