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.

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A352843 Expansion of e.g.f. exp(Sum_{k>=1} sigma_k(k) * x^k/k!).

Original entry on oeis.org

1, 1, 6, 44, 491, 6597, 110652, 2144606, 47988524, 1206275925, 33777572464, 1040200674416, 34967153135940, 1273241146218823, 49928549099500206, 2097300313258417056, 93953420539864844743, 4470694981375022862697, 225184078001798318202935
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2022

Keywords

Comments

Exponential transform of A023887.

Crossrefs

Cf. A295739, A274804, A352694.
Cf. A023887, A352841, A352842, A202477.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[E^(Sum[DivisorSigma[k, k]*x^k/k!, {k, 1, nmax}]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 15 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, sigma(k, k)*x^k/k!))))
  • PARI
    a(n) = if(n==0, 1, sum(k=1, n, sigma(k, k)*binomial(n-1, k-1)*a(n-k)));

Formula

a(0) = 1; a(n) = Sum_{k=1..n} sigma_k(k) * binomial(n-1,k-1) * a(n-k).

A353233 Möbius transform of sigma_n(n).

Original entry on oeis.org

1, 4, 27, 268, 3125, 47418, 823543, 16842736, 387440145, 10009763520, 285311670611, 8918294495628, 302875106592253, 11112685047823702, 437893920912783255, 18447025552964452096, 827240261886336764177, 39346558271491791438000, 1978419655660313589123979
Offset: 1

Views

Author

Wesley Ivan Hurt, Jun 24 2022

Keywords

Examples

			a(6) = 47418; a(6) = Sum_{d|6} sigma_d(d) * mu(6/d) = sigma_1(1) * mu(6/1) + sigma_2(2) * mu(6/2) + sigma_3(3) * mu(6/3) + sigma_6(6) * mu(6/6) = 1*1 + 5*(-1) + 28*(-1) + 47450*1 = 47418.

Links

Crossrefs

Cf. A008683 (mu), A023887.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, DivisorSigma[#, #]*MoebiusMu[n/#] &]; Array[a, 20] (* Wesley Ivan Hurt, Nov 12 2022 *)
  • PARI
    a(n) = sumdiv(n, d, sigma(d,d)*moebius(n/d)); \\ Michel Marcus, Jun 24 2022

Formula

a(n) = Sum_{d|n} sigma_d(d) * mu(n/d).

A356076 a(n) = Sum_{k=1..n} sigma_k(k) * floor(n/k).

Original entry on oeis.org

1, 7, 36, 315, 3442, 50926, 874471, 17717759, 405157961, 10414927743, 295726598356, 9214021189459, 312089127781714, 11424774177252514, 449318695090042077, 18896344248088180470, 846136606134424944649, 40192694877626991357901
Offset: 1

Views

Author

Seiichi Manyama, Jul 25 2022

Keywords

Links

Crossrefs

Partial sums of A344434.
Cf. A023887, A356046.

Programs

  • Mathematica
    Table[Sum[DivisorSigma[k,k]Floor[n/k],{k,n}],{n,20}] (* Harvey P. Dale, Sep 08 2024 *)
  • PARI
    a(n) = sum(k=1, n, sigma(k, k)*(n\k));
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, sigma(d, d)));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k, k)*x^k/(1-x^k))/(1-x))
  • Python
    from sympy import divisor_sigma
def A356079(n): return n+sum(divisor_sigma(k,k)*(n//k) for k in range(2,n+1)) # Chai Wah Wu, Jul 25 2022

Formula

a(n) = Sum_{k=1..n} Sum_{d|k} sigma_d(d).
G.f.: (1/(1-x)) * Sum_{k>=1} sigma_k(k) * x^k/(1 - x^k).
a(n) ~ n^n. - Vaclav Kotesovec, Aug 07 2022

A356587 Expansion of e.g.f. ( Product_{k>0} 1/(1 - (k * x)^k)^(1/k) )^x.

Original entry on oeis.org

1, 0, 2, 15, 236, 8490, 459234, 40325880, 4777773104, 767688946920, 156746202491880, 40056474754165320, 12448131138826294152, 4634982982962988690320, 2033625840922821008112144, 1039060311676326627685615800, 611331728108400284878223051520
Offset: 0

Views

Author

Seiichi Manyama, Aug 14 2022

Keywords

Crossrefs

Cf. A354623, A355064, A356554.
Cf. A023887, A356440, A356588.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-(k*x)^k)^(1/k))^x))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*sigma(j-1, j-1)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1, a(1) = 0; a(n) = Sum_{k=2..n} k! * sigma_{k-1}(k-1)/(k-1) * binomial(n-1,k-1) * a(n-k).

A376019 a(n) = Sum_{d|n} d^n * binomial(n/d-1,d-1).

Original entry on oeis.org

1, 1, 1, 17, 1, 129, 1, 769, 19684, 4097, 1, 1614804, 1, 98305, 86093443, 4295426049, 1, 3876302043, 1, 4398055948289, 156905298046, 41943041, 1, 2820680971922038, 298023223876953126, 805306369, 213516729579637, 1441151884248219649, 1
Offset: 1

Views

Author

Seiichi Manyama, Sep 06 2024

Keywords

Crossrefs

Cf. A023887, A376015.

Programs

  • PARI
    a(n) = sumdiv(n, d, d^n*binomial(n/d-1, d-1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, ((k*x)^k/(1-(k*x)^k))^k))
  • Python
    from math import comb
from itertools import takewhile
from sympy import divisors
def A376019(n): return sum(d**n*comb(n//d-1,d-1) for d in takewhile(lambda d:d**2<=n,divisors(n))) # Chai Wah Wu, Sep 06 2024

Formula

G.f.: Sum_{k>=1} ( (k*x)^k / (1 - (k*x)^k) )^k.
If p is prime, a(p) = 1.

A351750 a(n) = Sum_{p|n, p prime} p * sigma_p(p).

Original entry on oeis.org

0, 10, 84, 10, 15630, 94, 5764808, 10, 84, 15640, 3138428376732, 94, 3937376385699302, 5764818, 15714, 10, 14063084452067724991026, 94, 37589973457545958193355620, 15640, 5764892, 3138428376742, 480250763996501976790165756943064, 94, 15630, 3937376385699312
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 17 2022

Keywords

Comments

Inverse Möbius transform of n * sigma_n(n) * c(n), where c(n) is the characteristic function of primes (A010051). - Wesley Ivan Hurt, Apr 01 2025

Examples

			a(6) = 94; a(6) = Sum_{p|6, p prime} p * sigma_p(p) = 2 * sigma_2(2) + 3 * sigma_3(3) = 2 * (1^2 + 2^2) + 3 * (1^3 + 3^3) = 94.

Crossrefs

Cf. A010051, A023887 (sigma_n(n)), A351749.

Formula

a(n) = Sum_{d|n} d * sigma_d(d) * c(d), where c = A010051. - Wesley Ivan Hurt, Apr 01 2025

A351751 a(n) = Sum_{p|n, p prime} p^sigma_p(p).

Original entry on oeis.org

0, 32, 22876792454961, 32
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 17 2022

Keywords

Comments

a(5) has 2185 digits.
Inverse Möbius transform of n^sigma_n(n) * c(n), where c(n) is the characteristic function of primes (A010051). - Wesley Ivan Hurt, Apr 01 2025

Examples

			a(3) = 22876792454961; a(3) = Sum_{p|3, p prime} p^sigma_p(p) = 3^(1^3 + 3^3) = 3^28 = 22876792454961.

Crossrefs

Cf. A010051, A023887 (sigma_n(n)), A351749.

Formula

a(n) = Sum_{d|n} d^sigma_d(d) * c(d), where c = A010051. - Wesley Ivan Hurt, Apr 01 2025

A362627 Euler transform of sigma_n(n) (sum of n-th powers of divisors of n).

Original entry on oeis.org

1, 1, 6, 34, 322, 3588, 52844, 900082, 18111465, 411941506, 10548286788, 298667744593, 9286665651198, 314077164671106, 11484692279345752, 451291302965764596, 18966834595501974235, 848853415894558707472, 40305029983754331855502, 2023571200162099967806430, 107109031661019664234558776
Offset: 0

Views

Author

Wesley Ivan Hurt, Apr 28 2023

Keywords

Links

Crossrefs

Cf. A023887 (sigma_n(n)), A061256, A350503, A353233.

Programs

  • Mathematica
    a = Table[DivisorSigma[n, n], {n, 20}]; CoefficientList[Series[Product[1/(1 - x^m)^a[[m]], {m, 20}], {x, 0, 20}], x]

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^sigma(k,k).
a(n) = (1/n)*Sum_{k=1..n} a(n-k)*b(k) for n>1, a(0)=1, where b(k) = Sum_{d|k} d*sigma(d,d).
Previous Showing 51-58 of 58 results.

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