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 14 results. Next

A342629 a(n) = Sum_{d|n} (n/d)^(n-d).

Original entry on oeis.org

1, 3, 10, 69, 626, 7866, 117650, 2101265, 43047451, 1000390658, 25937424602, 743069105634, 23298085122482, 793728614541474, 29192926269590300, 1152925902670135553, 48661191875666868482, 2185913413229070900339, 104127350297911241532842
Offset: 1

Views

Author

Seiichi Manyama, Mar 16 2021

Keywords

Links

Crossrefs

Cf. A023887, A055225, A308668, A342628.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(n - #) &]; Array[a, 20] (* Amiram Eldar, Mar 17 2021 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(n-d));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^(k-1)*x^k/(1-k^(k-1)*x^k)))
  • Python
    from sympy import divisors
def A342629(n): return sum((n//d)**(n-d) for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 19 2022

Formula

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

A359112 a(n) = Sum_{d|n} (n/d) * d^(n-d).

Original entry on oeis.org

1, 3, 4, 13, 6, 109, 8, 777, 2197, 7541, 12, 374809, 14, 1675773, 31954096, 100794385, 18, 7391871271, 20, 163547770441, 2037381161992, 570634875581, 24, 1275177760626097, 476837158203151, 605750431288341, 450286447756825720, 2258377795760750777, 30
Offset: 1

Views

Author

Seiichi Manyama, Dec 17 2022

Keywords

Crossrefs

Cf. A000203, A167531.
Cf. A082245, A342628.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(n-#)*n/# &]; Array[a, 29] (* Amiram Eldar, Aug 27 2023 *)
  • PARI
    a(n) = sumdiv(n, d, n/d*d^(n-d));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(k*x)^k)^2))

Formula

G.f.: Sum_{k>=1} x^k/(1 - (k * x)^k)^2.
If p is prime, a(p) = 1 + p.

A354891 a(n) = n! * Sum_{d|n} d^(n - d) / d!.

Original entry on oeis.org

1, 3, 7, 73, 121, 9721, 5041, 1760641, 44452801, 562615201, 39916801, 3156125575681, 6227020801, 192873372531841, 222245415808416001, 14806216643368550401, 355687428096001, 34884164976924636172801, 121645100408832001
Offset: 1

Views

Author

Seiichi Manyama, Jun 10 2022

Keywords

Links

Crossrefs

Cf. A038507, A342628, A354888, A354890, A354893.

Programs

  • Mathematica
    a[n_] := n! * DivisorSum[n, #^(n - #)/#! &]; Array[a, 19] (* Amiram Eldar, Jun 10 2022 *)
  • PARI
    a(n) = n!*sumdiv(n, d, d^(n-d)/d!);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(k!*(1-(k*x)^k)))))

Formula

E.g.f.: Sum_{k>0} x^k/(k! * (1 - (k * x)^k)).
If p is prime, a(p) = 1 + p! = A038507(p).

A342675 a(n) = Sum_{d|n} d^(n-d+1).

Original entry on oeis.org

1, 3, 4, 13, 6, 120, 8, 1161, 2197, 16148, 12, 603190, 14, 5773008, 50422464, 201359377, 18, 16590656229, 20, 269768284118, 4748723771432, 3138430473896, 24, 2972582195034162, 476837158203151, 3937376419253748, 1350852564961601560, 4066515044181860654, 30, 1036488835382356683530, 32
Offset: 1

Views

Author

Seiichi Manyama, Mar 18 2021

Keywords

Links

Crossrefs

Cf. A023887, A294645, A342628, A342677.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(n - # + 1) &]; Array[a, 30] (* Amiram Eldar, Mar 18 2021 *)
  • PARI
    a(n) = sumdiv(n, d, d^(n-d+1));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, k*x^k/(1-(k*x)^k)))

Formula

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

A354893 a(n) = n! * Sum_{d|n} d^(n - d) / (n/d)!.

Original entry on oeis.org

1, 3, 7, 73, 121, 12361, 5041, 5308801, 44452801, 5681370241, 39916801, 16800125569921, 6227020801, 35897693762810881, 2134168822456070401, 190139202281277849601, 355687428096001, 3563095308471181273190401, 121645100408832001
Offset: 1

Views

Author

Seiichi Manyama, Jun 10 2022

Keywords

Links

Crossrefs

Cf. A038507, A342628, A354845, A354891, A354892.

Programs

  • Mathematica
    a[n_] := n! * DivisorSum[n, #^(n - #)/(n/#)! &]; Array[a, 19] (* Amiram Eldar, Jun 10 2022 *)
  • PARI
    a(n) = n!*sumdiv(n, d, d^(n-d)/(n/d)!);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, (exp((k*x)^k)-1)/k^k)))

Formula

E.g.f.: Sum_{k>0} (exp((k * x)^k) - 1)/k^k.
If p is prime, a(p) = 1 + p! = A038507(p).

A342607 a(n) = Sum_{d|n} phi(d)^(n-d).

Original entry on oeis.org

1, 2, 2, 3, 2, 11, 2, 19, 66, 1027, 2, 835, 2, 279939, 1052674, 69635, 2, 10114563, 2, 1074855939, 78364426242, 100000000003, 2, 4315152387, 1099511627778, 106993205379075, 101559973445634, 21937029021319171, 2, 1162183941554179, 2, 562950221856771, 10000000000001073741826
Offset: 1

Views

Author

Seiichi Manyama, Mar 16 2021

Keywords

Links

Crossrefs

Cf. A000010, A164941, A342608, A342612, A342628.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, EulerPhi[#]^(n - #) &]; Array[a, 30] (* Amiram Eldar, Mar 17 2021 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)^(n-d));
  • PARI
    a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^(n-n/gcd(k, n)-1));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(eulerphi(k)*x)^k)))

Formula

a(n) = Sum_{k=1..n} phi(n/gcd(k,n))^(n - n/gcd(k,n) - 1).
G.f.: Sum_{k>=1} x^k/(1 - (phi(k) * x)^k).
If p is prime, a(p) = 2.

A348146 a(n) = Sum_{d|n} (d!)^(n-d).

Original entry on oeis.org

1, 2, 2, 6, 2, 234, 2, 331842, 46658, 24883200258, 2, 139314179589392898, 2, 82606411253903523840004098, 619173642242176782338, 6984964247141514123665660725036072962, 2, 109110688415571335888754861121236891599318185050114, 2
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 02 2021

Keywords

Links

Crossrefs

Cf. A062363, A342628, A345465, A346196.

Programs

  • Mathematica
    Table[Sum[(i!)^(n - i) (1 - Ceiling[n/i] + Floor[n/i]), {i, n}], {n, 20}]
  • PARI
    a(n) = sumdiv(n, d, d!^(n-d)); \\ Seiichi Manyama, Oct 03 2021
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(k!*x)^k))) \\ Seiichi Manyama, Oct 03 2021

Formula

a(p) = 2 for primes p.
G.f.: Sum_{k>=1} x^k/(1 - (k! * x)^k). - Seiichi Manyama, Oct 03 2021

A356543 a(n) = Sum_{d|n} (d!)^(n/d-1).

Original entry on oeis.org

1, 2, 2, 4, 2, 12, 2, 34, 38, 138, 2, 1546, 2, 5106, 15698, 54274, 2, 889314, 2, 5689090, 25448258, 39917826, 2, 2486196610, 207360002, 6227024898, 131683574018, 215393466370, 2, 14769495662082, 2, 86475697160194, 1593350982706178, 355687428161538, 648227266560002
Offset: 1

Views

Author

Seiichi Manyama, Aug 11 2022

Keywords

Crossrefs

Cf. A087909, A342628, A356541.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (#)!^(n/# - 1) &]; Array[a, 35] (* Amiram Eldar, Aug 30 2023 *)
  • PARI
    a(n) = sumdiv(n, d, d!^(n/d-1));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-k!*x^k)))

Formula

G.f.: Sum_{k>=1} x^k/(1 - k! * x^k).
If p is prime, a(p) = 2.

A357051 a(n) = Sum_{d|n} 3^(n-d).

Original entry on oeis.org

1, 4, 10, 37, 82, 352, 730, 2998, 7291, 26488, 59050, 263170, 531442, 2127952, 5373460, 19669879, 43046722, 187086916, 387420490, 1607136634, 3878987860, 13947314752, 31381059610, 139902374692, 285916320883, 1129719740248, 2824682785300, 10460357985970
Offset: 1

Views

Author

Seiichi Manyama, Dec 20 2022

Keywords

Crossrefs

Cf. A074854, A112329, A359206.
Cf. A342628, A342629, A359203.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 3^(n-#) &]; Array[a, 28] (* Amiram Eldar, Aug 23 2023 *)
  • PARI
    a(n) = sumdiv(n, d, 3^(n-d));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, 3^(k-1)*x^k/(1-3^(k-1)*x^k)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(3*x)^k)))

Formula

G.f.: Sum_{k>=1} 3^(k-1) * x^k/(1 - 3^(k-1) * x^k).
G.f.: Sum_{k>=1} x^k/(1 - (3 * x)^k).

A363664 a(n) = Sum_{d|n} (n/d)^(n-n/d) * binomial(d+n-1,n).

Original entry on oeis.org

1, 4, 11, 56, 127, 1100, 1717, 19300, 64406, 383010, 352717, 23214660, 5200301, 191172406, 3465549077, 20859527460, 1166803111, 1010698826825, 17672631901, 102589250081802, 286539905316908, 75260204476154, 4116715363801, 548610025890719156
Offset: 1

Views

Author

Seiichi Manyama, Jun 14 2023

Keywords

Crossrefs

Cf. A342628, A359112, A363650.
Cf. A343548, A363662, A363663, A363667.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(n-n/#) * Binomial[# + n - 1, n] &]; Array[a, 25] (* Amiram Eldar, Jul 12 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(n-n/d)*binomial(d+n-1, n));

Formula

a(n) = [x^n] Sum_{k>0} x^k/(1 - (k*x)^k)^(n+1).
Showing 1-10 of 14 results. Next

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