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 1-9 of 9 results.

A350125 a(n) = Sum_{k=1..n} k^2 * floor(n/k)^n.

Original entry on oeis.org

1, 8, 40, 345, 3303, 50225, 833569, 17045934, 388654659, 10039636255, 285508661853, 8924967326015, 302927979357701, 11114722212099135, 437913155876193839, 18447871416712820782, 827249276230172525622, 39347009369000530723017
Offset: 1

Views

Author

Seiichi Manyama, Dec 15 2021

Keywords

Links

Crossrefs

Cf. A332469, A350109, A350123, A350124, A350128.

Programs

  • Mathematica
    a[n_] := Sum[k^2 * Floor[n/k]^n, {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Oct 04 2023 *)
  • PARI
    a(n) = sum(k=1, n, k^2*(n\k)^n);
  • PARI
    a(n) = sum(k=1, n, k^2*sumdiv(k, d, (d^n-(d-1)^n)/d^2));

Formula

a(n) = Sum_{k=1..n} k^2 * Sum_{d|k} (d^n - (d - 1)^n)/d^2.
a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} (k^n - (k - 1)^n) * x^k * (1 + x^k)/(1 - x^k)^3.
a(n) ~ n^n. - Vaclav Kotesovec, Dec 16 2021

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.

A350109 a(n) = Sum_{k=1..n} k * floor(n/k)^n.

Original entry on oeis.org

1, 6, 32, 295, 3201, 48321, 828323, 16910106, 388005909, 10019717653, 285409876785, 8920506515453, 302901435774351, 11113364096436947, 437903477186179875, 18447307498823123948, 827244767844150424228, 39346708569526147402819
Offset: 1

Views

Author

Seiichi Manyama, Dec 14 2021

Keywords

Links

Crossrefs

Main diagonal of A350106.
Cf. A319194, A332469.

Programs

  • Mathematica
    a[n_] := Sum[k * Floor[n/k]^n, {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Dec 14 2021 *)
  • PARI
    a(n) = sum(k=1, n, k*(n\k)^n);
  • PARI
    a(n) = sum(k=1, n, k*sumdiv(k, d, (d^n-(d-1)^n)/d));

Formula

a(n) = Sum_{k=1..n} k * Sum_{d|k} (d^n - (d - 1)^n)/d.
a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} (k^n - (k - 1)^n) * x^k/(1 - x^k)^2.
a(n) ~ n^n. - Vaclav Kotesovec, Dec 16 2021

A332468 a(n) = Sum_{k=1..n} mu(k) * floor(n/k)^n.

Original entry on oeis.org

1, 3, 25, 239, 3091, 45863, 821227, 16711423, 387138661, 9990174303, 285262663291, 8913906888703, 302861978789371, 11111328334033327, 437889112287422401, 18446462446101903615, 827238009323454485641, 39346257879101283645743, 1978418304199236175597105
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 13 2020

Keywords

Links

Crossrefs

Main diagonal of A344527.
Cf. A008683, A018805, A071778, A082540, A082544, A332469, A343978.

Programs

  • Magma
    [&+[MoebiusMu(k)*Floor(n/k)^n:k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 13 2020
  • Mathematica
    Table[Sum[MoebiusMu[k] Floor[n/k]^n, {k, 1, n}], {n, 1, 19}]
b[n_, k_] := b[n, k] = n^k - Sum[b[Floor[n/j], k], {j, 2, n}]; a[n_] := b[n, n]; Table[a[n], {n, 1, 19}]
  • PARI
    a(n)={sum(k=1, n, moebius(k) * floor(n/k)^n)} \\ Andrew Howroyd, Feb 13 2020
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A344527_T(n,k):
    if n == 0:
        return 0
    c, j, k1 = 1, 2, n//2
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A344527_T(k1,k)
        j, k1 = j2, n//j2
    return n*(n**(k-1)-1)-c+j
def A332468(n): return A344527_T(n,n) # Chai Wah Wu, Nov 02 2023

Formula

a(n) ~ n^n. - Vaclav Kotesovec, May 28 2021

A344724 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/k)^n.

Original entry on oeis.org

1, 3, 27, 240, 3094, 45990, 821484, 16711680, 387177517, 9990293423, 285263019633, 8913939911695, 302862111412779, 11111328866154037, 437889173336927557, 18446462747068745474, 827238010832411671962, 39346258082152478030126
Offset: 1

Views

Author

Seiichi Manyama, May 27 2021

Keywords

Links

Crossrefs

Main diagonal of A344726.
Cf. A332469.

Programs

  • Mathematica
    a[n_] := Sum[(-1)^(k + 1) * Quotient[n, k]^n, {k, 1, n}]; Array[a, 18] (* Amiram Eldar, May 27 2021 *)
  • PARI
    a(n) = sum(k=1, n, (-1)^(k+1)*(n\k)^n);
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*(d^n-(d-1)^n)));

Formula

a(n) = Sum_{k=1,..n} Sum_{d|k} (-1)^(k/d + 1) * (d^n - (d - 1)^n).
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, May 28 2021

A345176 a(n) = Sum_{k=1..n} floor(n/k)^k.

Original entry on oeis.org

1, 3, 5, 10, 12, 26, 28, 52, 73, 115, 117, 295, 297, 439, 713, 1160, 1162, 2448, 2450, 4644, 6832, 8902, 8904, 23536, 25639, 33857, 53247, 84961, 84963, 192237, 192239, 318477, 493909, 625015, 695789, 1761668, 1761670, 2285996, 3872598, 6255230, 6255232, 13392362
Offset: 1

Views

Author

Seiichi Manyama, Jun 10 2021

Keywords

Links

Crossrefs

Cf. A001923, A006218, A332469, A345098, A345100.

Programs

  • Mathematica
    a[n_] := Sum[Floor[n/k]^k, {k, 1, n}]; Array[a, 40] (* Amiram Eldar, Jun 10 2021 *)
  • PARI
    a(n) = sum(k=1, n, (n\k)^k);
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(j=1, N, (1-x^j)*sum(k=1, N, (k*x^k)^j))/(1-x))

Formula

G.f.: (1/(1 - x)) * Sum_{j>=1} Sum_{k>=1} (k*x^k)^j * (1 - x^j).
a(n) ~ 3^((n - mod(n,3))/3 + 1)/2. - Vaclav Kotesovec, Jun 11 2021

A347416 a(n) = Sum_{k=1..n} floor((n/k)^n).

Original entry on oeis.org

1, 5, 31, 276, 3238, 47463, 830415, 16845619, 388198577, 10009945747, 285452668383, 8918294580680, 302912273410475, 11112687415252836, 437907284782655738, 18447025981637731050, 827246579683710818081, 39346558272075085340201, 1978423430905859200399397
Offset: 1

Views

Author

Seiichi Manyama, Aug 31 2021

Keywords

Examples

			a(3) = [(3/1)^3] + [(3/2)^3] + [(3/3)^3] = 27 + 3 + 1 = 31.

Links

Crossrefs

Cf. A006218, A153818, A332469, A344675, A347415.

Programs

  • Mathematica
    a[n_] := Sum[Floor[(n/k)^n], {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Aug 31 2021 *)
  • PARI
    a(n) = sum(k=1, n, n^n\k^n);

Formula

a(n) ~ n^n. - Vaclav Kotesovec, Sep 14 2021

A356100 a(n) = Sum_{k=1..n} (k - 1)^n * floor(n/k).

Original entry on oeis.org

0, 1, 9, 99, 1301, 20581, 376891, 7914216, 186905206, 4915451602, 142368695176, 4506118905870, 154720069309364, 5729167232515112, 227585086051159866, 9654819212943764500, 435659280972794395356, 20836049921760968809231, 1052864549462731148832219
Offset: 1

Views

Author

Seiichi Manyama, Jul 26 2022

Keywords

Crossrefs

Cf. A121706, A236632, A319194, A332469.

Programs

  • Mathematica
    Table[Sum[(k-1)^n Floor[n/k],{k,n}],{n,20}] (* Harvey P. Dale, Dec 14 2024 *)
  • PARI
    a(n) = sum(k=1, n, (k-1)^n*(n\k));
  • PARI
    a(n) = sum(k=1, n, sigma(k, n)-(n\k)^n);
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, (d-1)^n));
  • Python
    def A356100(n): return sum((k-1)**n*(n//k) for k in range(2,n+1)) # Chai Wah Wu, Jul 26 2022

Formula

a(n) = A319194(n) - A332469(n).
a(n) = Sum_{k=1..n} Sum_{d|k} (d - 1)^n.
a(n) = [x^n] (1/(1-x)) * Sum_{k>=1} (k - 1)^n * x^k/(1 - x^k).

A332624 a(n) = Sum_{k=1..n} ceiling(n/k)^n.

Original entry on oeis.org

1, 5, 36, 289, 3433, 47578, 842499, 16850338, 389415029, 10010878371, 285679026506, 8918295095267, 302973286652448, 11112691430262573, 437929106387544254, 18447028378472722051, 827256956775203666857, 39346558275376372606086, 1978429667078835508142129
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 17 2020

Keywords

Crossrefs

Cf. A006590, A031971, A332469, A332490, A332623.

Programs

  • Magma
    [&+[Ceiling(n/k)^n:k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 17 2020
  • Mathematica
    Table[Sum[Ceiling[n/k]^n, {k, 1, n}], {n, 1, 19}]
Table[n + Sum[Sum[(d + 1)^n - d^n, {d, Divisors[k]}], {k, 1, n - 1}], {n, 1, 19}]
Table[SeriesCoefficient[x/(1 - x)^2 + x/(1 - x) Sum[((k + 1)^n - k^n) x^k/(1 - x^k), {k, 1, n}], {x, 0, n}], {n, 1, 19}]

Formula

a(n) = [x^n] x/(1 - x)^2 + (x/(1 - x)) * Sum_{k>=1} ((k + 1)^n - k^n) * x^k / (1 - x^k).
a(n) = n + Sum_{k=1..n-1} Sum_{d|k} ((d + 1)^n - d^n).
Showing 1-9 of 9 results.

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