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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A308676 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(d^k * n/d).

Original entry on oeis.org

1, 1, 3, 1, 5, 4, 1, 17, 28, 9, 1, 257, 19684, 273, 6, 1, 65537, 7625597484988, 4294967553, 3126, 24, 1, 4294967297, 443426488243037769948249630619149892804, 340282366920938463463374607431768276993, 298023223876953126, 47450, 8
Offset: 1

Views

Author

Seiichi Manyama, Jun 16 2019

Keywords

Examples

			Square array begins:
   1,    1,          1,                                       1, ...
   3,    5,         17,                                     257, ...
   4,   28,      19684,                           7625597484988, ...
   9,  273, 4294967553, 340282366920938463463374607431768276993, ...

Links

Crossrefs

Columns k=0..3 give A055225, A023887, A308670, A308675.
Cf. A308674.

Programs

  • Mathematica
    T[n_, k_] := DivisorSum[n, #^(n * #^(k-1)) &]; Table[T[k, n - k], {n, 1, 7}, {k, 1, n}] // Flatten (* Amiram Eldar, May 09 2021 *)

Formula

L.g.f. of column k: -log(Product_{j>=1} (1 - j^(j^k) * x^j)^(1/j)).

A321438 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^n.

Original entry on oeis.org

1, 3, 28, 239, 3126, 45990, 823544, 16711423, 387440173, 9990235398, 285311670612, 8913939907598, 302875106592254, 11111328602501550, 437893920912786408, 18446462594437808127, 827240261886336764178, 39346258082220810086373, 1978419655660313589123980, 104857499999905732078938574
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 09 2018

Keywords

Links

Crossrefs

Cf. A000593, A023887, A078306, A078307, A186633, A284900, A284926, A284927, A321385.

Programs

  • Magma
    m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[(k*x)^k/(1+(k*x)^k): k in [1..m]]) ));  // G. C. Greubel, Nov 11 2018
  • Mathematica
    Table[Sum[(-1)^(n/d + 1) d^n, {d, Divisors[n]}], {n, 20}]
nmax = 20; Rest[CoefficientList[Series[Sum[(k x)^k/(1 + (k x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 20; Rest[CoefficientList[Series[Log[Product[(1 + k^k x^k)^(1/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
  • PARI
    a(n) = sumdiv(n, d, (-1)^(n/d+1)*d^n); \\ Michel Marcus, Nov 09 2018

Formula

G.f.: Sum_{k>=1} (k*x)^k/(1 + (k*x)^k).
L.g.f.: log(Product_{k>=1} (1 + k^k*x^k)^(1/k)) = Sum_{n>=1} a(n)*x^n/n.
a(n) ~ n^n. - Vaclav Kotesovec, Nov 10 2018

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.

A344434 a(n) = Sum_{d|n} sigma_d(d), where sigma_k(n) is the sum of the k-th powers of the divisors of n.

Original entry on oeis.org

1, 6, 29, 279, 3127, 47484, 823545, 16843288, 387440202, 10009769782, 285311670613, 8918294591103, 302875106592255, 11112685049470800, 437893920912789563, 18447025552998138393, 827240261886336764179, 39346558271492566413252, 1978419655660313589123981
Offset: 1

Views

Author

Wesley Ivan Hurt, May 19 2021

Keywords

Comments

Inverse Möbius transform of sigma_n(n) (A023887). - Wesley Ivan Hurt, Mar 31 2025

Examples

			a(6) = Sum_{d|6} sigma_d(d) = (1^1) + (1^2 + 2^2) + (1^3 + 3^3) + (1^6 + 2^6 + 3^6 + 6^6) = 47484.

Links

Crossrefs

Cf. A023887 (sigma_n(n)), A245466, A321141, A334874, A343781.

Programs

  • Mathematica
    Table[Sum[DivisorSigma[k, k] (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 20}]
  • PARI
    a(n) = sumdiv(n, d, sigma(d, d)); \\ Michel Marcus, May 19 2021
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k, k)*x^k/(1-x^k))) \\ Seiichi Manyama, Jul 25 2022

Formula

If p is prime, a(p) = Sum_{d|p} sigma_d(d) = sigma_1(1) + sigma_p(p) = 1^1 + (1^p + p^p) = p^p + 2.
G.f.: Sum_{k>=1} sigma_k(k) * x^k/(1 - x^k). - Seiichi Manyama, Jul 25 2022

A064165 a(n) = tau( sigma_n(n) ), where tau is the number of divisors of n.

Original entry on oeis.org

1, 2, 6, 8, 8, 24, 16, 4, 8, 72, 96, 256, 32, 64, 1728, 64, 12, 384, 48, 2048, 3456, 128, 512, 2304, 256, 384, 2048, 9216, 128, 69120, 384, 2048, 184320, 2304, 81920, 2048, 128, 256, 9216, 1024, 64, 138240, 384, 16384, 32768, 3072, 2560, 131072, 64, 4194304, 196608
Offset: 1

Views

Author

Labos Elemer, Sep 19 2001

Keywords

Comments

Number of divisors of A023887(n).

Examples

			a(6) = 24; The sum of the 6th powers of the divisors of 6 is 1^6 + 2^6 + 3^6 + 6^6 = 47450, which has 24 divisors. - _Wesley Ivan Hurt_, May 04 2021

Links

Crossrefs

Cf. A000005 (tau), A000203 (sigma), A023887 (sigma_n(n)).

Programs

  • Mathematica
    Table[DivisorSigma[0,DivisorSigma[w,w]],{w,30}] (* Harvey P. Dale, Jul 08 2019 *)
  • PARI
    a(n) = numdiv(sigma(n, n)); \\ Michel Marcus, May 05 2021
  • Python
    from math import prod
from collections import Counter
from sympy import factorint
def A064165(n): return prod(r+1 for q,r in sum((Counter(factorint((p**(n*(e+1))-1)//(p**n-1))) for p, e in factorint(n).items()),Counter()).items()) # Chai Wah Wu, Jan 28 2022

Extensions

More terms from Wesley Ivan Hurt, May 04 2021

A308569 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*n).

Original entry on oeis.org

1, 1, 2, 1, 5, 2, 1, 17, 28, 3, 1, 65, 730, 273, 2, 1, 257, 19684, 65793, 3126, 4, 1, 1025, 531442, 16781313, 9765626, 47450, 2, 1, 4097, 14348908, 4295032833, 30517578126, 2177317874, 823544, 4, 1, 16385, 387420490, 1099512676353, 95367431640626, 101560344351050, 678223072850, 16843009, 3
Offset: 1

Views

Author

Seiichi Manyama, Jun 08 2019

Keywords

Examples

			Square array begins:
   1,    1,       1,           1,              1, ...
   2,    5,      17,          65,            257, ...
   2,   28,     730,       19684,         531442, ...
   3,  273,   65793,    16781313,     4295032833, ...
   2, 3126, 9765626, 30517578126, 95367431640626, ...

Links

Crossrefs

Columns k=0..2 give A000005, A023887, A308570.
Rows n=1..2 give A000012, A052539.
A(n,n) gives A308571.
Cf. A308504, A308509.

Programs

  • Mathematica
    T[n_, k_] := DivisorSum[n, #^(k*n) &]; Table[T[k, n - k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, May 11 2021 *)
  • PARI
    T(n,k) = sumdiv(n, d, d^(k*n));
matrix(5, 5, n, k, T(n,k-1)) \\ Michel Marcus, Jun 08 2019

Formula

L.g.f. of column k: -log(Product_{j>=1} (1 - (j^k*x)^j)^(1/j)).

A344060 a(n) = Sum_{d|n} sigma(d)^n.

Original entry on oeis.org

1, 10, 65, 2483, 7777, 2990810, 2097153, 2568661988, 10604761518, 3570527751850, 743008370689, 232227195048256531, 793714773254145, 21035724521219881850, 504857283427304833025, 727429690188773950335429, 2185911559738696531969, 43567528891100073055151954340, 5242880000000000000000001
Offset: 1

Views

Author

Seiichi Manyama, May 08 2021

Keywords

Crossrefs

Cf. A007429, A023887, A065018, A342471, A344044, A344047, A344061.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, DivisorSigma[1 , #]^n &]; Array[a, 19] (* Amiram Eldar, May 08 2021 *)
  • PARI
    a(n) = sumdiv(n, d, sigma(d)^n);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (sigma(k)*x)^k/(1-(sigma(k)*x)^k)))

Formula

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

A346196 a(n) = Sum_{d|n} (d!)^n.

Original entry on oeis.org

1, 5, 217, 331793, 24883200001, 139314069504046721, 82606411253903523840000001, 6984964247141514123629140487675314433, 109110688415571316480344899355894085582848010077697, 395940866122425193243875570782668457763038823019173642240000001025
Offset: 1

Views

Author

Seiichi Manyama, Jul 10 2021

Keywords

Links

Crossrefs

Cf. A023887, A036739, A217576, A345465, A348146.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (#!)^n &]; Array[a, 10] (* Amiram Eldar, Aug 30 2023 *)
  • PARI
    a(n) = sumdiv(n, d, d!^n);
  • PARI
    my(N=20, 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!)^p.

A356437 a(n) = n! * Sum_{k=1..n} sigma_k(k)/k.

Original entry on oeis.org

1, 7, 77, 1946, 84754, 6202524, 636369348, 89979720144, 16431405256656, 3796658174518560, 1077102230236529760, 368915006390671969920, 149873555740938949215360, 71297150722148582901815040, 39244301012876892023553235200
Offset: 1

Views

Author

Seiichi Manyama, Aug 07 2022

Keywords

Crossrefs

Cf. A023887, A356297, A356436, A356440.

Programs

  • Mathematica
    Table[n! * Sum[DivisorSigma[k, k]/k, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 07 2022 *)
  • PARI
    a(n) = n!*sum(k=1, n, sigma(k, k)/k);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, log(1-(k*x)^k)/k)/(1-x)))

Formula

E.g.f.: -(1/(1-x)) * Sum_{k>0} log(1 - (k*x)^k)/k.
a(n) ~ n! * n^(n-1). - Vaclav Kotesovec, Aug 07 2022

A179504 a(n) = sigma(2n,n) for n>0 with a(0)=1.

Original entry on oeis.org

1, 3, 21, 252, 4369, 103158, 3037530, 106237176, 4311810305, 198756808749, 10250010815226, 584603613083988, 36529334450076658, 2481455748310337022, 182070232515259616850, 14349345894391097803752
Offset: 0

Views

Author

Paul D. Hanna, Sep 21 2010

Keywords

Comments

sigma(n,k) denotes the sum of the k-th powers of the divisors of n.

Links

Programs

  • Mathematica
    Table[If[n == 0, 1, DivisorSigma[n, 2 n]], {n, 0, 15}] (* Michael De Vlieger, Jul 12 2017 *)
  • PARI
    a(n)=if(n==0,1,sigma(2*n,n))
  • Python
    from sympy import divisor_sigma
def A179504(n): return divisor_sigma(n<<1,n) if n else 1 # Chai Wah Wu, Jul 08 2022

Formula

After initial term, equals the logarithmic derivative of A156360.
Conjecture: a(n) = A023887(n)*((2^n - 1)/(2^(n*(A007814(n) + 1)) - 1) + 2^n) for n > 0. - Velin Yanev, Jul 08 2017
a(n) ~ (2*n)^n. - Vaclav Kotesovec, Aug 21 2017
Previous Showing 21-30 of 58 results. Next

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