A166375 -id:A166375 - cp's OEIS frontend

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

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

A166373 Triangle read by rows for floor(j^2 / n) with n >= 2 and 1<=j

Original entry on oeis.org

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

Offset: 2

Christopher Hunt Gribble, Oct 13 2009. Offset corrected Oct 18 2009

			Part of the triangle from which the sequence is constructed is shown below.
.....j..1..2..3..4..5..6..7..8..9.10.11.12.13.14
...n
...2....0
...3....0..1
...4....0..1..2
...5....0..0..1..3
...6....0..0..1..2..4
...7....0..0..1..2..3..5
...8....0..0..1..2..3..4..6
...9....0..0..1..1..2..4..5..7
..10....0..0..0..1..2..3..4..6..8
..11....0..0..0..1..2..3..4..5..7..9
..12....0..0..0..1..2..3..4..5..6..8.10
..13....0..0..0..1..1..2..3..4..6..7..9.11
..14....0..0..0..1..1..2..3..4..5..7..8.10.12
..15....0..0..0..1..1..2..3..4..5..6..8..9.11.13

A165974 is contained in this sequence.

Cf. A166381 (column sums), A166375 (row sums).

T(n,j)=j^2\n \\ Charles R Greathouse IV, Jan 16 2012

A165993 a(n) = sum_{j=1..prime(n)-1} floor (j^2/prime(n)).

Original entry on oeis.org

0, 1, 4, 11, 31, 44, 80, 103, 157, 252, 293, 420, 520, 575, 695, 884, 1105, 1180, 1431, 1617, 1704, 2007, 2217, 2552, 3040, 3300, 3439, 3713, 3852, 4144, 5255, 5595, 6120, 6305, 7252, 7457, 8060, 8695, 9141, 9804, 10507, 10740, 11983, 12224, 12740

Offset: 1

Christopher Hunt Gribble, Oct 03 2009

nonn

C. H. Gribble, Table of n, a(n) for n=1..78498 (i.e. for primes < 10^6).

Cf. A166375, A165974, A014817.

Table[Sum[Floor[j^2/n],{j,n-1}],{n,Prime[Range[50]]}] (* Harvey P. Dale, Aug 10 2014 *)

a(n) = sum(j=1, prime(n)-1, floor (j^2/prime(n))) \\ Michel Marcus, Jun 20 2013

a(n)=my(p=prime(n));sum(j=1,p-1,j^2\p) \\ Charles R Greathouse IV, Jun 20 2013

a(n) = A166375(prime(n)-1). - Charles R Greathouse IV, Jun 28 2013

Definition rephrased by R. J. Mathar, Oct 09 2009

A166387 a(n) = sum (floor (j^2/n), 1 <= j <= n-1) - floor ((n-1)(n-2)/3), n >= 2.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 2, 2, 0, 1, 3, 0, 1, 3, 4, 0, 2, 1, 3, 2, 1, 3, 6, 2, 0, 3, 3, 0, 3, 3, 6, 2, 0, 3, 7, 0, 1, 5, 6, 0, 2, 1, 5, 4, 3, 5, 10, 4, 2, 3, 3, 0, 3, 5, 8, 2, 0, 3, 9, 0, 3, 7, 10, 0, 2, 1, 5, 4, 3, 7, 12, 0, 0, 7, 5, 2, 5, 5, 12, 6, 0, 3, 9, 0, 1, 7, 8, 0, 4, 3, 7, 4, 5, 9, 16, 0, 4

Offset: 2

Christopher Hunt Gribble, Oct 13 2009

nonn

a(n) = 0 when n = 2, any prime of the form 4k+1 with k >= 1 and any product of these without repetition, e.g. 2x5x17.

G. C. Greubel, Table of n, a(n) for n = 2..10000

Cf. A166375, A128422.

Table[Sum[Floor[k^2/n], {k, 1, n - 1}] - Floor[(n - 1)*(n - 2)/3], {n, 2, 100}] (* G. C. Greubel, May 12 2016 *)

a(n) = sum(j=1,n-1, j^2\n) - ((n-1)*(n-2))\3 \\ Michel Marcus, Jun 21 2013

a(n) = A166375(n) - A128422(n).

Corrected and enhanced by Christopher Hunt Gribble, Dec 01 2009

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

A166373 Triangle read by rows for floor(j^2 / n) with n >= 2 and 1<=j

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

A165993 a(n) = sum_{j=1..prime(n)-1} floor (j^2/prime(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A166387 a(n) = sum (floor (j^2/n), 1 <= j <= n-1) - floor ((n-1)(n-2)/3), n >= 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions