cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

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

Original entry on oeis.org

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

Views

Author

Benoit Cloitre, Feb 24 2013

Keywords

Comments

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

References

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

Links

Crossrefs

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).

Programs

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

Formula

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

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

Views

Author

Ilya Gutkovskiy, Feb 13 2020

Keywords

Links

Crossrefs

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

Programs

  • Magma
    [&+[Floor(n/k)^n:k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 13 2020
  • Mathematica
    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}]
  • PARI
    a(n)={sum(k=1, n, floor(n/k)^n)} \\ Andrew Howroyd, Feb 13 2020
  • Python
    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

Formula

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

Views

Author

Vaclav Kotesovec, Sep 02 2018

Keywords

Links

Crossrefs

Cf. A000005, A000203, A024916, A064602, A064603, A064604.
Cf. A006218, A222548, A318742, A318744.

Programs

  • Magma
    [&+[Floor(n/k)^4:k in [1..n]]:n in [1..36]]; // Marius A. Burtea, Jul 16 2019
  • Mathematica
    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}]]
  • PARI
    a(n) = sum(k=1, n, (n\k)^4); \\ Michel Marcus, Sep 03 2018
  • Python
    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

Formula

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

A318744 a(n) = Sum_{k=1..n} floor(n/k)^5.

Original entry on oeis.org

1, 33, 245, 1058, 3160, 8054, 17086, 33860, 60353, 103437, 164489, 257945, 380407, 556001, 779865, 1085840, 1457122, 1958008, 2544540, 3312306, 4205650, 5336264, 6618976, 8254674, 10059777, 12298021, 14792045, 17829881, 21130663, 25189011, 29518163, 34749419
Offset: 1

Views

Author

Vaclav Kotesovec, Sep 02 2018

Keywords

Links

Crossrefs

Cf. A000005, A000203, A024916, A064602, A064603, A064604.
Cf. A006218, A222548, A318742, A318743.

Programs

  • Mathematica
    Table[Sum[Floor[n/k]^5, {k, 1, n}], {n, 1, 40}]
Accumulate[Table[DivisorSigma[0, k] - 5*DivisorSigma[1, k] + 10*DivisorSigma[2, k] - 10*DivisorSigma[3, k] + 5*DivisorSigma[4, k], {k, 1, 40}]]
  • PARI
    a(n) = sum(k=1, n, (n\k)^5); \\ Michel Marcus, Sep 03 2018
  • PARI
    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)) \\ Seiichi Manyama, May 27 2021
  • Python
    from math import isqrt
def A318744(n): return -(s:=isqrt(n))**6+sum((q:=n//k)*(k**5-(k-1)**5+q**4) for k in range(1,s+1)) # Chai Wah Wu, Oct 26 2023

Formula

a(n) = A006218(n) - 5*A024916(n) + 10*A064602(n) - 10*A064603(n) + 5*A064604(n).
a(n) ~ zeta(5) * n^5.

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

Views

Author

Seiichi Manyama, May 27 2021

Keywords

Examples

			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, ...

Links

Crossrefs

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

Programs

  • Mathematica
    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);
  • PARI
    T(n, k) = sum(j=1, n, sumdiv(j, d, d^k-(d-1)^k));
  • Python
    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

Formula

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.

A344721 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/k)^3.

Original entry on oeis.org

1, 7, 27, 56, 118, 196, 324, 448, 685, 901, 1233, 1549, 2019, 2445, 3157, 3664, 4482, 5262, 6290, 7128, 8536, 9598, 11118, 12392, 14255, 15743, 18087, 19711, 22149, 24417, 27209, 29251, 32771, 35327, 39087, 42048, 46046, 49244, 54180, 57512, 62434, 66838, 72258, 76246
Offset: 1

Views

Author

Seiichi Manyama, May 27 2021

Keywords

Links

Crossrefs

Column k=3 of A344726.
Cf. A318742.

Programs

  • Mathematica
    a[n_] := Sum[(-1)^(k + 1) * Quotient[n, k]^3, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 27 2021 *)
Accumulate[Table[-3*DivisorSigma[0, n] + 2*DivisorSigma[0, 2*n] + 6*DivisorSigma[1, n] - 3*DivisorSigma[1, 2*n] - 9/2 * DivisorSigma[2, n] + 3/2 * DivisorSigma[2, 2*n], {n, 1, 50}]] (* Vaclav Kotesovec, May 28 2021 *)
  • PARI
    a(n) = sum(k=1, n, (-1)^(k+1)*(n\k)^3);
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*(d^3-(d-1)^3)));
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k/(1+x^k))/(1-x))

Formula

a(n) = Sum_{k=1,..n} Sum_{d|k} (-1)^(k/d + 1) * (d^3 - (d - 1)^3).
G.f.: (1/(1 - x)) * Sum_{k>=1} (k^3 - (k - 1)^3) * x^k/(1 + x^k).
a(n) ~ 3*zeta(3)*n^3/4. - Vaclav Kotesovec, May 28 2021

A350108 a(n) = Sum_{k=1..n} k * floor(n/k)^3.

Original entry on oeis.org

1, 10, 32, 87, 153, 309, 443, 722, 1005, 1443, 1785, 2605, 3087, 3951, 4875, 6154, 6988, 8809, 9855, 12057, 13853, 16001, 17543, 21347, 23478, 26484, 29440, 33696, 36162, 41994, 44816, 50351, 54755, 59909, 64577, 73524, 77558, 84002, 90142, 100072, 105034
Offset: 1

Views

Author

Seiichi Manyama, Dec 14 2021

Keywords

Links

Crossrefs

Column 3 of A350106.
Cf. A024916, A143128, A143127, A318742.

Programs

  • Mathematica
    a[n_] := Sum[k * Floor[n/k]^3, {k, 1, n}]; Array[a, 40] (* Amiram Eldar, Dec 14 2021 *)
Accumulate[Table[(1 + 3*k)*DivisorSigma[1, k] - 3*k*DivisorSigma[0, k], {k, 1, 50}]] (* Vaclav Kotesovec, Dec 16 2021 *)
  • PARI
    a(n) = sum(k=1, n, k*(n\k)^3);
  • PARI
    a(n) = sum(k=1, n, k*sumdiv(k, d, (d^3-(d-1)^3)/d));
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k/(1-x^k)^2)/(1-x))
  • Python
    from math import isqrt
def A350108(n): return -(s:=isqrt(n))**4*(s+1)+sum((q:=n//k)*(k**2*(3*(q+1))+k*(q*((q<<1)-3)-3)+q+1) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 31 2023

Formula

a(n) = Sum_{k=1..n} k * Sum_{d|k} (d^3 - (d - 1)^3)/d.
G.f.: (1/(1 - x)) * Sum_{k>=1} (k^3 - (k - 1)^3) * x^k/(1 - x^k)^2.
From Vaclav Kotesovec, Aug 03 2022: (Start)
a(n) = A024916(n) + 3*A143128(n) - 3*A143127(n).
a(n) ~ Pi^2*n^3/6 - 3*n^2*log(n)/2. (End)

A350124 a(n) = Sum_{k=1..n} k^2 * floor(n/k)^3.

Original entry on oeis.org

1, 12, 40, 121, 207, 473, 649, 1142, 1611, 2401, 2853, 4647, 5285, 6879, 8759, 11452, 12558, 16739, 18127, 23353, 27129, 31171, 33219, 43573, 47524, 53210, 59538, 69996, 73274, 89694, 93446, 107195, 116731, 126545, 137505, 164580, 169946, 182244, 195644, 225454
Offset: 1

Views

Author

Seiichi Manyama, Dec 15 2021

Keywords

Links

Crossrefs

Cf. A318742, A350108, A350123, A350125.
Cf. A064602, A143128, A319085.

Programs

  • Mathematica
    Accumulate[Table[DivisorSigma[2, k] - 3*k*DivisorSigma[1, k] + 3*k^2*DivisorSigma[0, k], {k, 1, 50}]] (* Vaclav Kotesovec, Dec 17 2021 *)
  • PARI
    a(n) = sum(k=1, n, k^2*(n\k)^3);
  • PARI
    a(n) = sum(k=1, n, k^2*sumdiv(k, d, (d^3-(d-1)^3)/d^2));
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k*(1+x^k)/(1-x^k)^3)/(1-x))
  • Python
    from math import isqrt
def A350124(n): return (-(s:=isqrt(n))**4*(s+1)*(2*s+1) + sum((q:=n//k)*(k*(3*(k-1))+q*(k*(9*(k-1))+q*(k*(12*k-6)+2)+3)+1) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 21 2023

Formula

a(n) = Sum_{k=1..n} k^2 * Sum_{d|k} (d^3 - (d - 1)^3)/d^2.
G.f.: (1/(1 - x)) * Sum_{k>=1} (k^3 - (k - 1)^3) * x^k * (1 + x^k)/(1 - x^k)^3.
From Vaclav Kotesovec, Aug 03 2022: (Start)
a(n) = A064602(n) - 3*A143128(n) + 3*A319085(n).
a(n) ~ n^3 * (log(n) + 2*gamma + (zeta(3) - 1)/3 - Pi^2/6), where gamma is the Euler-Mascheroni constant A001620. (End)

A356249 a(n) = Sum_{k=1..n} (k * floor(n/k))^3.

Original entry on oeis.org

1, 16, 62, 219, 405, 1053, 1523, 2948, 4407, 7041, 8703, 15283, 17949, 24657, 32685, 44806, 50536, 70687, 78573, 105411, 125879, 149879, 163565, 222425, 247476, 286134, 327634, 396258, 423084, 532236, 564818, 664763, 738095, 821693, 904937, 1107618, 1162268, 1277588, 1395760
Offset: 1

Views

Author

Seiichi Manyama, Jul 31 2022

Keywords

Links

Crossrefs

Column k=3 of A356250.
Cf. A000537, A024916, A318742, A350123.
Cf. A064603, A356125, A319086.

Programs

  • Mathematica
    a[n_] := Sum[(k * Floor[n/k])^3, {k, 1, n}]; Array[a, 40] (* Amiram Eldar, Jul 31 2022 *)
  • PARI
    a(n) = sum(k=1, n, (k*(n\k))^3);
  • PARI
    a(n) = sum(k=1, n, k^3*sumdiv(k, d, 1-(1-1/d)^3));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k*(1+4*x^k+x^(2*k))/(1-x^k)^4)/(1-x))
  • Python
    from math import isqrt
def A356249(n): return -(s:=isqrt(n))**5*(s+1)**2 + sum((q:=n//k)**2*(k*(3*(k-1))+q*(k*(k*(4*k+6)-6)+q*(k*(3*(k-1))+1)+2)+1) for k in range(1,s+1))>>2 # Chai Wah Wu, Oct 21 2023

Formula

a(n) = Sum_{k=1..n} k^3 * Sum_{d|k} (1 - (1 - 1/d)^3).
G.f.: (1/(1 - x)) * Sum_{k>=1} (k^3 - (k - 1)^3) * x^k * (1 + 4*x^k + x^(2*k))/(1 - x^k)^4.
From Vaclav Kotesovec, Aug 02 2022: (Start)
a(n) = A064603(n) - 3*A356125(n) + 3*A319086(n).
a(n) ~ n^4 * (Pi^2/8 + Pi^4/360 - 3*zeta(3)/4). (End)

A358877 Triangle read by rows: T(n,k) is the number of cubes of side length k that can be placed inside a cube of side length n without overlap, 1 <= k <= n.

Original entry on oeis.org

1, 8, 1, 27, 1, 1, 64, 8, 1, 1, 125, 8, 1, 1, 1, 216, 27, 8, 1, 1, 1, 343, 27, 8, 1, 1, 1, 1, 512, 64, 8, 8, 1, 1, 1, 1, 729, 64, 27, 8, 1, 1, 1, 1, 1, 1000, 125, 27, 8, 8, 1, 1, 1, 1, 1, 1331, 125, 27, 8, 8, 1, 1, 1, 1, 1, 1, 1728, 216, 64, 27, 8, 8, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Torlach Rush, Feb 17 2023

Keywords

Comments

T(n,k) is a cube 1 <= k <= n.
Alternative construction: Write each column k with each cube repeated k times.
Row sums of triangle are A318742.

Examples

			Triangle begins:
   1;
   8,   1;
  27,   1,  1;
  64,   8,  1,  1;
 125,   8,  1,  1,  1;
 216,  27,  8,  1,  1,  1;
 343,  27,  8,  1,  1,  1, 1;
 512,  64,  8,  8,  1,  1, 1, 1;
 729,  64, 27,  8,  1,  1, 1, 1, 1;
1000, 125, 27,  8,  8,  1, 1, 1, 1, 1;
1331, 125, 27,  8,  8,  1, 1, 1, 1, 1, 1;
1728, 216, 64, 27,  8,  8, 1, 1, 1, 1, 1, 1;
...

Crossrefs

Cf. A000578, A318742, A360610.

Programs

  • Python
    def T(n, k): return (n//k)**3
Showing 1-10 of 10 results.

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