A165993 -id:A165993 - cp's OEIS frontend

A165974 Irregular triangle read by rows: floor(j^2 / p) for p prime and j = 1 to p-1.

0, 0, 1, 0, 0, 1, 3, 0, 0, 1, 2, 3, 5, 0, 0, 0, 1, 2, 3, 4, 5, 7, 9, 0, 0, 0, 1, 1, 2, 3, 4, 6, 7, 9, 11, 0, 0, 0, 0, 1, 2, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 15, 17, 0, 0, 0, 0, 1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 17, 19, 21

Offset: 1

Christopher Hunt Gribble, Oct 02 2009

The irregular triangle of numbers is:
....j..1..2..3..4..5..6..7..8..9.10.11.12.13.14.15.16.17.18.19.20.21.22
..p
..2....0
..3....0..1
..5....0..0..1..3
..7....0..0..1..2..3..5
.11....0..0..0..1..2..3..4..5..7..9
.13....0..0..0..1..1..2..3..4..6..7..9.11
.17....0..0..0..0..1..2..2..3..4..5..7..8..9.11.13.15
.19....0..0..0..0..1..1..2..3..4..5..6..7..8.10.11.13.15.17
.23....0..0..0..0..1..1..2..2..3..4..5..6..7..8..9.11.12.14.15.17.19.21

C. H. Gribble, Flattened irregular triangle read by rows of primes < 300

Cf. A165993 (row sums).

Keyword:tabf added by R. J. Mathar, Oct 09 2009

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

Original entry on oeis.org

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

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A165974 Irregular triangle read by rows: floor(j^2 / p) for p prime and j = 1 to p-1.

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

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