A014817 -id:A014817 - cp's OEIS frontend

A227842 First differences of A014817.

1, 2, 3, 2, 4, 5, 6, 5, 5, 8, 9, 6, 10, 11, 12, 7, 13, 12, 15, 12, 14, 17, 18, 13, 15, 20, 19, 16, 22, 21, 24, 17, 21, 26, 27, 18, 26, 29, 28, 21, 29, 28, 33, 28, 30, 33, 36, 27, 31, 34, 35, 32, 38, 39, 40, 31, 37, 42, 45, 32, 44, 45, 46, 33, 45, 44, 49, 44, 46, 51, 52, 37, 49, 56, 49, 48, 54, 53, 60, 47

Offset: 1

N. J. A. Sloane, Aug 08 2013

nonn

Cf. A014817, A227841.

A227842 := proc(n)
    A014817(n+1)-A014817(n) ;
end proc:
seq(A227842(n),n=1..80) ; # R. J. Mathar, Aug 09 2013

Differences[Table[Sum[Floor[k^2/n],{k,n}],{n,100}]] (* Harvey P. Dale, Sep 13 2024 *)

A227841 Partial sums of A014817.

Original entry on oeis.org

1, 3, 7, 14, 23, 36, 54, 78, 107, 141, 183, 234, 291, 358, 436, 526, 623, 733, 855, 992, 1141, 1304, 1484, 1682, 1893, 2119, 2365, 2630, 2911, 3214, 3538, 3886, 4251, 4637, 5049, 5488, 5945, 6428, 6940, 7480, 8041, 8631, 9249, 9900, 10579, 11288, 12030

Offset: 1

N. J. A. Sloane, Aug 08 2013

nonn

Cf. A014817, A227842.

Accumulate[Table[Sum[Floor[k^2/n],{k,n}],{n,50}]] (* Harvey P. Dale, Sep 27 2023 *)

A166375 a(n) = sum (floor (j^2/n)) taken over 1 <= j <= n-1.

Original entry on oeis.org

0, 1, 3, 4, 7, 11, 16, 20, 24, 31, 39, 44, 53, 63, 74, 80, 92, 103, 117, 128, 141, 157, 174, 186, 200, 219, 237, 252, 273, 293, 316, 332, 352, 377, 403, 420, 445, 473, 500, 520, 548, 575, 607, 634, 663, 695, 730, 756, 786, 819, 853, 884, 921, 959, 998

Offset: 2

Christopher Hunt Gribble, Oct 13 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 2..1000

Row sums of A166373. Cf. A014817.

A166375 := proc(n)
    add( floor(j^2/n),j=1..n-1) ;
end proc: # R. J. Mathar, Jul 21 2015

Table[Sum[Floor[k^2/n], {k, 1, n - 1}], {n, 2, 100}] (* G. C. Greubel, May 10 2016 *)

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

a(n) = A014817(n) - n. - R. J. Mathar, Jul 21 2015

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

A177041 Sum(round(k^2/n),k=1..n).

Original entry on oeis.org

1, 3, 4, 7, 11, 16, 20, 26, 31, 39, 44, 53, 63, 74, 82, 94, 105, 119, 128, 141, 157, 174, 188, 204, 221, 239, 254, 275, 295, 318, 336, 360, 377, 403, 422, 447, 475, 502, 526, 554, 581, 611, 636, 665, 697, 732, 760, 794, 825, 861

Offset: 1

Mircea Merca, Dec 09 2010

The round function, also called the nearest integer function, is defined here by round(x)=floor(x+1/2).

Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.

Cf. A014817.

A177041 := proc(n)
    add( round(j^2/n),j=1..n) ;
end proc:

Table[Sum[Floor[k^2/n + 1/2], {k, n}], {n, 50}]

A014818 a(n) is the sum over all floor(k^3/n), k=0 to n inclusive.

Original entry on oeis.org

1, 4, 11, 24, 43, 71, 109, 160, 222, 298, 391, 502, 631, 781, 953, 1150, 1369, 1617, 1891, 2196, 2531, 2899, 3301, 3740, 4215, 4726, 5283, 5874, 6511, 7193, 7921, 8700, 9521, 10396, 11323, 12306, 13339, 14431, 15581, 16792, 18061, 19394, 20791, 22254, 23784

Offset: 1

N. J. A. Sloane

nonn

M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkhauser, 1985, p. 103.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A014817, A014819.

[&+[Floor(k^3/n): k in [1..n]]: n in [1..50]]; // Vincenzo Librandi, Feb 12 2017

Maple
```
f := m->sum( floor(k^3 / m), k=0..m);
```

Table[Sum[Floor[k^3 / n], {k, n}], {n, 50}] (* Vincenzo Librandi, Feb 12 2017 *)

a(n) = sum(k=0, n, k^3\n); \\ Michel Marcus, Feb 12 2017

A014819 a(n) = Sum_{k=1..n} floor(k^4/n).

Original entry on oeis.org

1, 8, 32, 88, 195, 377, 666, 1096, 1701, 2530, 3630, 5056, 6863, 9115, 11884, 15240, 19249, 24012, 29606, 36126, 43665, 52327, 62220, 73452, 86137, 100398, 116364, 134158, 153915, 175789, 199908, 226432, 255501, 287288, 321958, 359672, 400599, 444927, 492842

Offset: 1

N. J. A. Sloane

nonn

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..1000

Cf. A014817, A014818.

[(&+[Floor(k^4/n): k in [1..n]]): n in [1..40]]; // G. C. Greubel, Nov 21 2018

f := m-> add( floor((nu)^4/m), nu=0..m): seq(f(n), n=1..40);

Table[Sum[Floor[k^4/n], {k, 1, n}], {n, 1, 40}] (* G. C. Greubel, Nov 21 2018 *)

vector(40, n, sum(k=1,n, floor(k^4/n))) \\ G. C. Greubel, Nov 21 2018

[sum(floor(k^4/n) for k in (1..n)) for n in (1..40)] # G. C. Greubel, Nov 21 2018

Title improved by Sean A. Irvine, Nov 21 2018

A175908 3*sum(k=1..n, floor(k^2/n)) - n^2.

Original entry on oeis.org

2, 2, 3, 5, 2, 3, 5, 8, 6, 2, 5, 9, 2, 5, 9, 14, 2, 6, 5, 11, 6, 5, 11, 18, 8, 2, 9, 11, 2, 9, 11, 20, 6, 2, 11, 21, 2, 5, 15, 20, 2, 6, 5, 17, 12, 11, 17, 30, 14, 8, 9, 11, 2, 9, 17, 26, 6, 2, 11, 27, 2, 11, 21, 32, 2, 6, 5, 17, 12, 11, 23, 36, 2, 2, 21, 17, 8, 15, 17, 38

Offset: 1

John W. Layman, Oct 14 2010

nonn

According to the reference, a(p*q) = a(p) + a(q) - 2 whenever p and q are distinct primes with p congruent to q modulo 4.

The sequences of indices n where a(n)=2 is {1, 2, 5, 10, 13, 17, 26, ...}, which appears to be A020893 (squarefree sums of two squares). This has been confirmed for the first 500 terms. [John W. Layman, May 16 2011]

Walter Blumberg, Problem 11529, Amer. Math. Monthly, 117 (2010), 742.

Cf. A020893.

A175908:=n->3*add(floor(k^2/n),k=1..n)-n^2: seq(A175908(n), n=1..60); # Wesley Ivan Hurt, Jul 10 2014

Table[3 Sum[Floor[k^2/n], {k, n}] - n^2, {n, 60}] (* Wesley Ivan Hurt, Jul 10 2014 *)

a(n) = 3*sum(k=1, n, k^2\n) - n^2; \\ Michel Marcus, Jul 09 2014

a(n) = 3*A014817(n) - A000290(n). - Wesley Ivan Hurt, Jul 10 2014

A014785 a(n) = Sum_{0<=k<=n} ceiling(k^2/n).

Original entry on oeis.org

1, 3, 6, 9, 13, 18, 24, 30, 35, 43, 52, 61, 69, 80, 92, 102, 113, 125, 140, 155, 169, 184, 202, 220, 231, 251, 270, 291, 309, 332, 354, 376, 397, 419, 446, 469, 493, 520, 550, 578, 601, 631, 660, 693, 721, 754, 788

Offset: 1

N. J. A. Sloane

nonn

M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkhauser, 1985, p. 103.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A014817 (with floor), A177041 (with round).

[&+[Ceiling(k^2/n):k in [0..n]]:n in [1..50]]; // Marius A. Burtea, Dec 31 2019

Maple
```
f := n->sum( ceil(k^2/n), k=0..n);
```

Table[Sum[Ceiling[k^2/n],{k,0,n}],{n,50}] (* Harvey P. Dale, Oct 24 2013 *)

cp's OEIS Frontend

A227842 First differences of A014817.

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A227841 Partial sums of A014817.

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A166375 a(n) = sum (floor (j^2/n)) taken over 1 <= j <= n-1.

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

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A177041 Sum(round(k^2/n),k=1..n).

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A014818 a(n) is the sum over all floor(k^3/n), k=0 to n inclusive.

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A014819 a(n) = Sum_{k=1..n} floor(k^4/n).

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A175908 3*sum(k=1..n, floor(k^2/n)) - n^2.

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