A318744 -id:A318744 - cp's OEIS frontend

A222548 a(n) = Sum_{k=1..n} floor(n/k)^2.

1, 5, 11, 22, 32, 52, 66, 92, 115, 147, 169, 219, 245, 289, 333, 390, 424, 496, 534, 612, 672, 740, 786, 898, 957, 1037, 1113, 1219, 1277, 1413, 1475, 1595, 1687, 1791, 1883, 2056, 2130, 2246, 2354, 2526, 2608, 2792, 2878, 3040, 3190, 3330, 3424, 3662, 3773

Offset: 1

Benoit Cloitre, Feb 24 2013

nonn

a(n) is the number of common divisors of integers 1<=i,j<=n over all ordered pairs (i,j). - Geoffrey Critzer, Jan 15 2015

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 98.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
MathOverflow, The Second Moment of a Sum of Floor Functions, 2012.

Cf. A013661, A018806, A024916, A153818.

Similar sequences for Sum_{k=1..n} floor(n/k)^m: A006218 (m=1), this sequence (m=2), A318742 (m=3), A318743 (m=4), A318744 (m=5).

[&+[Floor(n/k)^2:k in [1..n] ]: n in [1..40]]; // Marius A. Burtea, Jul 16 2019

Table[Sum[Floor[n/k]^2, {k, n}], {n, 50}] (* T. D. Noe, Feb 26 2013 *)
Table[nn = n;Total[Level[Table[Table[DivisorSigma[0, GCD[i, j]], {i, 1, nn}], {j, 1, nn}], {2}]], {n, 1, 49}] (* Geoffrey Critzer, Jan 15 2015 *)
Table[Sum[2*DivisorSigma[1, k] - DivisorSigma[0, k], {k, 1, n}], {n, 1, 50}] (* Vaclav Kotesovec, Sep 02 2018 *)

PARI
```
a(n)=sum(k=1,n,(n\k)^2)
```

from math import isqrt
def A222548(n): return -(s:=isqrt(n))**3 + sum((q:=n//k)*((k<<1)+q-1) for k in range(1,s+1)) # Chai Wah Wu, Oct 21 2023

a(n) = zeta(2)*n^2 + O(n log n).
a(n) = 2*A024916(n) - A006218(n). - Vaclav Kotesovec, Sep 02 2018
G.f.: (1/(1 - x)) * Sum_{k>=1} (2*k - 1) * x^k/(1 - x^k). - Ilya Gutkovskiy, Jul 16 2019
a(n) = Sum_{d=1..n} (2*d-1)*floor(n/d). [Uspensky and Heaslet] - Michael Somos, Feb 16 2020
a(n) = Sum_{k=1..n} Sum_{d|k} floor(n/d). - Ridouane Oudra, Jul 16 2020
a(n) = Sum_{i=1..n} Sum_{j=1..n} tau(gcd(i,j)). - Ridouane Oudra, Nov 23 2021

A318742 a(n) = Sum_{k=1..n} floor(n/k)^3.

Original entry on oeis.org

1, 9, 29, 74, 136, 254, 382, 596, 833, 1173, 1505, 2057, 2527, 3209, 3921, 4856, 5674, 6928, 7956, 9474, 10882, 12608, 14128, 16506, 18369, 20797, 23141, 26129, 28567, 32259, 35051, 38963, 42483, 46675, 50435, 55904, 59902, 65156, 70092, 76460

Offset: 1

Vaclav Kotesovec, Sep 02 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Richard J. Mathar, Chebyshev approximation of x^m (-log x)^l in the interval 0 <= x <= 1, arXiv:2408.15212 [math.CA], 2024. See p. 2.

Cf. A000005, A000203, A024916, A064602, A064603, A064604.

Cf. A006218, A222548, A318743, A318744.

[&+[Floor(n/k)^3:k in [1..n]]: n in [1..40]]; // Marius A. Burtea, Jul 16 2019

Table[Sum[Floor[n/k]^3, {k, 1, n}], {n, 1, 40}]
Accumulate[Table[DivisorSigma[0, k] - 3*DivisorSigma[1, k] + 3*DivisorSigma[2, k], {k, 1, 40}]]

a(n) = sum(k=1, n, (n\k)^3); \\ Michel Marcus, Sep 03 2018

from math import isqrt
def A318742(n): return -(s:=isqrt(n))**4 + sum((q:=n//k)*(3*k*(k-1)+q**2+1) for k in range(1,s+1)) # Chai Wah Wu, Oct 21 2023

a(n) = A006218(n) - 3*A024916(n) + 3*A064602(n).
a(n) ~ zeta(3) * n^3.
G.f.: (1/(1 - x)) * Sum_{k>=1} (3*k*(k - 1) + 1) * x^k/(1 - x^k). - Ilya Gutkovskiy, Jul 16 2019

A332469 a(n) = Sum_{k=1..n} floor(n/k)^n.

Original entry on oeis.org

1, 5, 29, 274, 3160, 47452, 825862, 16843268, 387702833, 10009826727, 285360679985, 8918294547447, 302888236005847, 11112685321898449, 437898668488710801, 18447025705612363530, 827242514466399305122, 39346558271561286347116, 1978421007121668206129316

Offset: 1

Ilya Gutkovskiy, Feb 13 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A006218, A222548, A318742, A318743, A318744, A319194, A332468.

[&+[Floor(n/k)^n:k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 13 2020

Table[Sum[Floor[n/k]^n, {k, 1, n}], {n, 1, 19}]
Table[SeriesCoefficient[1/(1 - x) Sum[(k^n - (k - 1)^n) x^k/(1 - x^k), {k, 1, n}], {x, 0, n}], {n, 1, 19}]

a(n)={sum(k=1, n, floor(n/k)^n)} \\ Andrew Howroyd, Feb 13 2020

from math import isqrt
def A332469(n): return -(s:=isqrt(n))**(n+1)+sum((q:=n//k)*(k**n-(k-1)**n+q**(n-1)) for k in range(1,s+1)) # Chai Wah Wu, Oct 26 2023

a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} (k^n - (k - 1)^n) * x^k / (1 - x^k).

a(n) ~ n^n. - Vaclav Kotesovec, Jun 11 2021

A318743 a(n) = Sum_{k=1..n} floor(n/k)^4.

Original entry on oeis.org

1, 17, 83, 274, 644, 1396, 2502, 4388, 6919, 10743, 15385, 22407, 30233, 41209, 53853, 70650, 88636, 113308, 138654, 172332, 207984, 252416, 298002, 358654, 417873, 492065, 569061, 663427, 756053, 875541, 989063, 1130915, 1272967, 1441383, 1607147, 1817080

Offset: 1

Vaclav Kotesovec, Sep 02 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A024916, A064602, A064603, A064604.

Cf. A006218, A222548, A318742, A318744.

[&+[Floor(n/k)^4:k in [1..n]]:n in [1..36]]; // Marius A. Burtea, Jul 16 2019

Table[Sum[Floor[n/k]^4, {k, 1, n}], {n, 1, 40}]
Accumulate[Table[-DivisorSigma[0, k] + 4*DivisorSigma[1, k] - 6*DivisorSigma[2, k] + 4*DivisorSigma[3, k], {k, 1, 40}]]

a(n) = sum(k=1, n, (n\k)^4); \\ Michel Marcus, Sep 03 2018

from math import isqrt
def A318743(n): return -(s:=isqrt(n))**5+sum((q:=n//k)*(k**4-(k-1)**4+q**3) for k in range(1,s+1)) # Chai Wah Wu, Oct 26 2023

a(n) = -A006218(n) + 4*A024916(n) - 6*A064602(n) + 4*A064603(n).
a(n) ~ zeta(4) * n^4.
a(n) ~ Pi^4 * n^4 / 90.
G.f.: (1/(1 - x)) * Sum_{k>=1} (2*k - 1) * (2*k^2 - 2*k + 1) * x^k/(1 - x^k). - Ilya Gutkovskiy, Jul 16 2019

A344725 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} floor(n/j)^k.

Original entry on oeis.org

1, 1, 3, 1, 5, 5, 1, 9, 11, 8, 1, 17, 29, 22, 10, 1, 33, 83, 74, 32, 14, 1, 65, 245, 274, 136, 52, 16, 1, 129, 731, 1058, 644, 254, 66, 20, 1, 257, 2189, 4162, 3160, 1396, 382, 92, 23, 1, 513, 6563, 16514, 15692, 8054, 2502, 596, 115, 27, 1, 1025, 19685, 65794, 78256, 47452, 17086, 4388, 833, 147, 29

Offset: 1

Seiichi Manyama, May 27 2021

			Square array begins:
   1,  1,   1,    1,    1,     1, ...
   3,  5,   9,   17,   33,    65, ...
   5, 11,  29,   83,  245,   731, ...
   8, 22,  74,  274, 1058,  4162, ...
  10, 32, 136,  644, 3160, 15692, ...
  14, 52, 254, 1396, 8054, 47452, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=1..5 give A006218, A222548, A318742, A318743, A318744.
T(n,n) gives A332469.
Cf. A306533, A344726.

T[n_, k_] := Sum[Quotient[n, j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, May 27 2021 *)

PARI
```
T(n, k) = sum(j=1, n, (n\j)^k);
```

T(n, k) = sum(j=1, n, sumdiv(j, d, d^k-(d-1)^k));

from math import isqrt
from itertools import count, islice
def A344725_T(n,k): return -(s:=isqrt(n))**(k+1)+sum((q:=n//w)*(w**k-(w-1)**k+q**(k-1)) for w in range(1,s+1))
def A344725_gen(): # generator of terms
     return (A344725_T(k+1,n-k) for n in count(1) for k in range(n))
A344725_list = list(islice(A344725_gen(),30)) # Chai Wah Wu, Oct 26 2023

G.f. of column k: (1/(1 - x)) * Sum_{j>=1} (j^k - (j - 1)^k) * x^j/(1 - x^j).

T(n,k) = Sum_{j=1..n} Sum_{d|j} d^k - (d - 1)^k.

A344723 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/k)^5.

Original entry on oeis.org

1, 31, 243, 992, 3094, 7564, 16596, 31744, 58237, 97117, 158169, 241837, 364299, 521829, 745693, 1018120, 1389402, 1837302, 2423834, 3105432, 3998776, 5007286, 6289998, 7738784, 9543887, 11537207, 14031231, 16717879, 20018661, 23629281, 27958433, 32577739, 38219963, 44148743

Offset: 1

Seiichi Manyama, May 27 2021

nonn

In general, for m > 1, Sum_{k=1..n} (-1)^(k+1) * floor(n/k)^m ~ (1 - 2^(1-m)) * zeta(m) * n^m. - Vaclav Kotesovec, May 28 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Column k=5 of A344726.

Cf. A318744.

a[n_] := Sum[(-1)^(k + 1) * Quotient[n, k]^5, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 27 2021 *)
Accumulate[Table[-3*DivisorSigma[0, n] + 2*DivisorSigma[0, 2*n] + 10*DivisorSigma[1, n] - 5*DivisorSigma[1, 2*n] - 15*DivisorSigma[2, n] + 5*DivisorSigma[2, 2*n] + 25/2 * DivisorSigma[3, n] - 5/2 * DivisorSigma[3, 2*n] - 45/8 *DivisorSigma[4, n] + 5/8 * DivisorSigma[4, 2*n], {n, 1, 50}]] (* Vaclav Kotesovec, May 28 2021 *)

a(n) = sum(k=1, n, (-1)^(k+1)*(n\k)^5);

a(n) = sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*(d^5-(d-1)^5)));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (k^5-(k-1)^5)*x^k/(1+x^k))/(1-x))

a(n) = Sum_{k=1,..n} Sum_{d|k} (-1)^(k/d + 1) * (d^5 - (d - 1)^5).
G.f.: (1/(1 - x)) * Sum_{k>=1} (k^5 - (k - 1)^5) * x^k/(1 + x^k).
a(n) ~ 15*zeta(5)*n^5/16. - Vaclav Kotesovec, May 28 2021

cp's OEIS Frontend

A222548 a(n) = Sum_{k=1..n} floor(n/k)^2.

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A318742 a(n) = Sum_{k=1..n} floor(n/k)^3.

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A332469 a(n) = Sum_{k=1..n} floor(n/k)^n.

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A318743 a(n) = Sum_{k=1..n} floor(n/k)^4.

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A344725 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} floor(n/j)^k.

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A344723 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/k)^5.

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