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 16 results. Next

A332533 a(n) = (1/n) * Sum_{k=1..n} floor(n/k) * n^k.

Original entry on oeis.org

1, 4, 15, 92, 790, 9384, 137326, 2397352, 48428487, 1111122360, 28531183329, 810554859732, 25239592620853, 854769763924104, 31278135039463245, 1229782938533709200, 51702516368332126932, 2314494592676172411516, 109912203092257573556274, 5518821052632117898282620
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 16 2020

Keywords

Links

Crossrefs

Cf. A024916, A268235, A308313, A308814, A319194, A320095.
Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), A344818 (q=-3), A344817 (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), this sequence (q=n).

Programs

  • Magma
    A332533:= func< n | (&+[Floor(n/j)*n^(j-1): j in [1..n]]) >;
[A332533(n): n in [1..40]]; // G. C. Greubel, Jun 27 2024
  • Maple
    seq(add(n^(k-1)*floor(n/k), k=1..n), n=1..60); # Ridouane Oudra, Mar 05 2023
  • Mathematica
    Table[(1/n) Sum[Floor[n/k] n^k, {k, 1, n}], {n, 1, 20}]
Table[(1/n) Sum[Sum[n^d, {d, Divisors[k]}], {k, 1, n}], {n, 1, 20}]
Table[SeriesCoefficient[(1/(1 - x)) Sum[x^k/(1 - n x^k), {k, 1, n}], {x, 0, n}], {n, 1, 20}]
  • PARI
    a(n) = sum(k=1, n, (n\k)*n^k)/n; \\ Michel Marcus, Feb 16 2020
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, n^(d-1))); \\ Seiichi Manyama, May 29 2021
  • SageMath
    def A332533(n): return sum((n//j)*n^(j-1) for j in range(1,n+1))
[A332533(n) for n in range(1,41)] # G. C. Greubel, Jun 27 2024

Formula

a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} x^k / (1 - n*x^k).
a(n) = (1/n) * Sum_{k=1..n} Sum_{d|k} n^d.
a(n) ~ n^(n-1). - Vaclav Kotesovec, May 28 2021
a(n) = (1/(n-1)) * Sum_{k=1..n} (n^floor(n/k) - 1), for n>=2. - Ridouane Oudra, Mar 05 2023

A321141 a(n) = Sum_{d|n} sigma_n(d).

Original entry on oeis.org

1, 6, 29, 291, 3127, 48246, 823545, 16909060, 387459858, 10019533302, 285311670613, 8920489178073, 302875106592255, 11113363271736486, 437893951444713443, 18447307036548136965, 827240261886336764179, 39346708467688595378892, 1978419655660313589123981
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 28 2018

Keywords

Links

Crossrefs

Cf. A000005, A007429, A007433, A023887, A319194, A320940, A321140.

Programs

  • Magma
    [&+[DivisorSigma(n, d):d in Divisors(n)]:n in [1..20]]; // Vincenzo Librandi, Feb 16 2020
  • Maple
    with(numtheory): seq(coeff(series(add(sigma[n](k)*x^k/(1-x^k),k=1..n),x,n+1), x, n), n = 1 .. 20); # Muniru A Asiru, Oct 28 2018
  • Mathematica
    Table[Sum[DivisorSigma[n, d], {d, Divisors[n]}] , {n, 19}]
Table[SeriesCoefficient[Sum[DivisorSigma[n, k] x^k/(1 - x^k), {k, 1, n}], {x, 0, n}], {n, 19}]
  • PARI
    a(n) = sumdiv(n, d, sigma(d, n)); \\ Michel Marcus, Oct 28 2018
  • Python
    from sympy import divisor_sigma, divisors
def A321141(n):
    return sum(divisor_sigma(d,0)*(n//d)**n for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020

Formula

a(n) = [x^n] Sum_{k>=1} sigma_n(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} d^n*tau(n/d).
a(n) ~ n^n. - Vaclav Kotesovec, Feb 16 2020

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

A319649 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=1..n} j^k * floor(n/j).

Original entry on oeis.org

1, 1, 3, 1, 4, 5, 1, 6, 8, 8, 1, 10, 16, 15, 10, 1, 18, 38, 37, 21, 14, 1, 34, 100, 111, 63, 33, 16, 1, 66, 278, 373, 237, 113, 41, 20, 1, 130, 796, 1335, 999, 489, 163, 56, 23, 1, 258, 2318, 4957, 4461, 2393, 833, 248, 69, 27, 1, 514, 6820, 18831, 20583, 12513, 4795, 1418, 339, 87, 29
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 09 2018

Keywords

Examples

			Square array begins:
   1,   1,    1,    1,     1,      1,  ...
   3,   4,    6,   10,    18,     34,  ...
   5,   8,   16,   38,   100,    278,  ...
   8,  15,   37,  111,   373,   1335,  ...
  10,  21,   63,  237,   999,   4461,  ...
  14,  33,  113,  489,  2393,  12513,  ...

Links

Crossrefs

Columns k=0..5 give A006218, A024916, A064602, A064603, A064604, A248076.
Cf. A082771, A109974, A319194 (diagonal).

Programs

  • Mathematica
    Table[Function[k, Sum[j^k Floor[n/j] , {j, 1, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[j^k x^j/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, Sum[DivisorSigma[k, j], {j, 1, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
  • Python
    from itertools import count, islice
from math import isqrt
from sympy import bernoulli
def A319649_T(n,k): return (((s:=isqrt(n))+1)*(bernoulli(k+1)-bernoulli(k+1,s+1))+sum(w**k*(k+1)*((q:=n//w)+1)-bernoulli(k+1)+bernoulli(k+1,q+1) for w in range(1,s+1)))//(k+1) + int(k==0)
def A319649_gen(): # generator of terms
     return (A319649_T(k+1,n-k-1) for n in count(1) for k in range(n))
A319649_list = list(islice(A319649_gen(),30)) # Chai Wah Wu, Oct 24 2023

Formula

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

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

A350128 a(n) = Sum_{k=1..n} k^n * floor(n/k)^2.

Original entry on oeis.org

1, 8, 44, 417, 4545, 69905, 1207937, 24904806, 575256641, 14947281595, 427836523971, 13429362462839, 457637290140469, 16843379604615375, 665494379869134005, 28102480944522059434, 1262906802939553227382, 60182948301301262753877
Offset: 1

Views

Author

Seiichi Manyama, Dec 15 2021

Keywords

Links

Crossrefs

Cf. A222548, A319194, A350107, A350125.

Programs

  • Mathematica
    Table[Sum[k^n Floor[n/k]^2,{k,n}],{n,20}] (* Harvey P. Dale, Feb 11 2022 *)
  • PARI
    a(n) = sum(k=1, n, k^n*(n\k)^2);
  • PARI
    a(n) = sum(k=1, n, 2*k*sigma(k, n-1)-sigma(k, n));
  • Python
    from math import isqrt
from sympy import bernoulli
def A350128(n): return (((s:=isqrt(n))+1)*(1-s)*(bernoulli(n+1,s+1)-(b:=bernoulli(n+1)))+sum(k**n*(n+1)*(((q:=n//k)+1)*(q-1))+(1-2*k)*(b-bernoulli(n+1,q+1)) for k in range(1,s+1)))//(n+1) # Chai Wah Wu, Oct 21 2023

Formula

a(n) = Sum_{k=1..n} 2 * k * sigma_{n-1}(k) - sigma_{n}(k).
a(n) ~ n^n / (1 - exp(-1)). - Vaclav Kotesovec, Dec 16 2021

A356239 a(n) = Sum_{k=1..n} k^n * sigma_0(k).

Original entry on oeis.org

1, 9, 71, 963, 9873, 231749, 2976863, 86348423, 1824883450, 55584932826, 1104642697680, 64932555347084, 1366828157222090, 61273696016238014, 2581786206601959958, 129797968403021602450, 3678372903755436314440, 295835829367866540495396
Offset: 1

Views

Author

Seiichi Manyama, Jul 30 2022

Keywords

Links

Crossrefs

Cf. A000005, A319194, A356129, A356243.

Programs

  • Maple
    f:= proc(n) local k; add(k^n * numtheory:-tau(k),k=1..n) end proc:
map(f, [$1..30]); # Robert Israel, Jan 21 2024
  • Mathematica
    a[n_] := Sum[k^n * DivisorSigma[0, k], {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Jul 30 2022 *)
  • PARI
    a(n) = sum(k=1, n, k^n*sigma(k, 0));
  • PARI
    a(n) = sum(k=1, n, k^n*sum(j=1, n\k, j^n));
  • Python
    from math import isqrt
from sympy import bernoulli
def A356239(n): return (-(bernoulli(n+1, (s:=isqrt(n))+1)-(b:=bernoulli(n+1)))**2//(n+1) + sum(k**n*(bernoulli(n+1, n//k+1)-b)<<1 for k in range(1,s+1)))//(n+1) # Chai Wah Wu, Oct 21 2023

Formula

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

A065805 a(n) = Sum_{j=0..n} sigma_j(n).

Original entry on oeis.org

2, 10, 44, 377, 3912, 57214, 960808, 19261862, 435877584, 11123320200, 313842837684, 9729290348250, 328114698808288, 11967567841654610, 469172063576559648, 19676848703371278527, 878942778254232811956, 41661071646298278566892, 2088331858752553232964220
Offset: 1

Views

Author

Labos Elemer, Nov 21 2001

Keywords

Examples

			For n = 6, a(6) = 4 + 12 + 50 + 252 + 1394 + 8052 + 47450 = 57214.

Links

Crossrefs

Cf. A000005, A000203, A001157, A319194.

Programs

  • Mathematica
    a[n_] := Apply[Plus, Table[DivisorSigma[w, n], {w, 0, n}]]; Array[a, 30]
a[n_] := DivisorSum[n, (#^(n + 1) - 1)/(# - 1) &, # > 1 &] + n + 1; Array[a, 30] (* Amiram Eldar, Mar 01 2025 *)
  • PARI
    a(n) = sum(j=0, n, sigma(n, j)); \\ Harry J. Smith, Oct 31 2009
  • PARI
    a(n) = sumdiv(n, d, if(d == 1, n+1, (d^(n+1) - 1)/(d - 1))); \\ Amiram Eldar, Mar 01 2025

Formula

a(n) ~ n^(n+1) / (n-1). - Vaclav Kotesovec, Sep 11 2018
a(n) = n + 1 + Sum_{d|n, dAmiram Eldar, Mar 02 2025

A308313 a(n) = Sum_{k=1..n} (-1)^(n-k) * k^n * floor(n/k).

Original entry on oeis.org

1, 2, 22, 203, 2285, 33855, 609345, 12420372, 284964519, 7347342215, 209807114169, 6554034238459, 222469737401739, 8159109186320903, 321461264348047819, 13538455640979049698, 606976994365011212414, 28864017965496692865925, 1451086990386146504580735, 76896033641977171208887465
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 22 2019

Keywords

Crossrefs

Cf. A319194, A319649.

Programs

  • Mathematica
    Table[Sum[(-1)^(n - k) k^n Floor[n/k] , {k, 1, n}], {n, 1, 20}]
Table[SeriesCoefficient[1/(1 + x) Sum[k^n x^k/(1 - (-x)^k), {k, 1, n}], {x, 0, n}], {n, 1, 20}]
Table[(-1)^n Sum[DivisorSigma[n, k] - 2 Total[Select[Divisors[k], OddQ]^n], {k, 1, n}], {n, 1, 20}]
  • PARI
    a(n)={sum(k=1, n, (-1)^(n-k) * k^n * (n\k))} \\ Andrew Howroyd, Aug 22 2019
  • Python
    from math import isqrt
from sympy import bernoulli
def A308313(n): return (-1 if n&1 else 1)*((((s:=isqrt(m:=n>>1))+1)*(bernoulli(n+1)-bernoulli(n+1,s+1))<

Formula

a(n) = [x^n] (1/(1 + x)) * Sum_{k>=1} k^n * x^k/(1 - (-x)^k).
a(n) = Sum_{k=1..n} Sum_{d|k} (-1)^(n-d) * d^n.
a(n) ~ c * n^n, where c = 1/(1 + exp(-1)) = 0.7310585786300048792511592418218362743651446401650565192763659... - Vaclav Kotesovec, Aug 22 2019, updated Jul 19 2021
Let A(n,k) = Sum_{j=1..n} j^k * floor(n/j). Then a(n) = (-1)^n*(2^(n+1)*A(floor(n/2),n)-A(n,n)). - Chai Wah Wu, Oct 28 2023

A355887 a(n) = Sum_{k=1..n} k^k * floor(n/k).

Original entry on oeis.org

1, 6, 34, 295, 3421, 50109, 873653, 17651130, 405071647, 10405074777, 295716745389, 9211817240589, 312086923832843, 11424093750214407, 449317984131076935, 18896062057857406028, 846136323944194170206, 40192544399241119212807
Offset: 1

Views

Author

Seiichi Manyama, Jul 20 2022

Keywords

Links

Crossrefs

Partial sums of A062796.
Cf. A006218, A024916, A064602, A064603, A064604, A248076, A319194.

Programs

  • PARI
    a(n) = sum(k=1, n, n\k*k^k);
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, d^d));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x)^k/(1-x^k))/(1-x))
  • Python
    def A355887(n): return n*(1+n**(n-1))+sum(k**k*(n//k) for k in range(2,n)) if n>1 else 1 # Chai Wah Wu, Jul 21 2022

Formula

a(n) = Sum_{k=1..n} Sum_{d|k} d^d.
G.f.: (1/(1-x)) * Sum_{k>0} (k * x)^k/(1 - x^k).
Showing 1-10 of 16 results. Next

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