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 1-5 of 5 results.

A321259 a(n) = sigma_n(n) - n^n.

Original entry on oeis.org

0, 1, 1, 17, 1, 794, 1, 65793, 19684, 9766650, 1, 2194095090, 1, 678223089234, 30531927033, 281479271743489, 1, 150196195641350171, 1, 100000096466944316978, 558545874543637211, 81402749386839765307626, 1, 79501574308536809523296482, 298023223876953126
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 01 2018

Keywords

Comments

a(n) is the sum of n-th powers of proper divisors of n.

Links

Crossrefs

Cf. A000312, A001065, A023887, A067558, A276634, A279363, A279364, A292919, A321258.

Programs

  • Magma
    [DivisorSigma(n, n) - n^n: n in [1..30]]; // Vincenzo Librandi, Nov 02 2018
  • Mathematica
    Table[DivisorSigma[n, n] - n^n, {n, 25}]
nmax = 25; Rest[CoefficientList[Series[Sum[(k x)^(2 k)/(1 - (k x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
  • PARI
    a(n) = sigma(n, n) - n^n; \\ Michel Marcus, Nov 02 2018

Formula

G.f.: Sum_{k>=1} (k*x)^(2*k)/(1 - (k*x)^k).
a(n) = A023887(n) - A000312(n).
a(n) = A321258(n,n).
a(n) = 1 if n is prime.

A321260 a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^(sigma_n(k)-k^n).

Original entry on oeis.org

1, 0, 1, 1, 18, 2, 861, 132, 106024, 40910, 72980055, 6838271, 228282942581, 27620223647, 2050169324675668, 352809815149813, 87174966874755673105, 6798293425492905407, 18318448554980083512011863, 1187839217207171380193247, 11258918803635775614062752424535
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 01 2018

Keywords

Crossrefs

Cf. A283333, A318783, A318784, A319647, A321258, A321261.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[1/(1 - x^k)^(DivisorSigma[n, k] - k^n), {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[Exp[Sum[DivisorSigma[n + 1, k] x^(2 k)/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 20}]

Formula

a(n) = [x^n] exp(Sum_{k>=1} sigma_(n+1)(k)*x^(2*k)/(k*(1 - x^k))).

A321261 a(n) = [x^n] Product_{k>=1} (1 + x^k)^(sigma_n(k)-k^n).

Original entry on oeis.org

1, 0, 1, 1, 17, 2, 859, 131, 105508, 40907, 72916903, 6834168, 228239366293, 27616985835, 2050004858009336, 352807044193881, 87173272463714343166, 6798224808203572198, 18318379579349549499397403, 1187836799227050499295342, 11258903016282277676462826232428
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 01 2018

Keywords

Crossrefs

Cf. A281268, A318844, A319107, A321042, A321258, A321260.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1 + x^k)^(DivisorSigma[n, k] - k^n), {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[Exp[Sum[Sum[(-1)^(k/d + 1) d (DivisorSigma[n, d] - d^n), {d, Divisors[k]}] x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 20}]

Formula

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

A322080 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{p|n, p prime} p^k.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 0, 4, 3, 1, 0, 8, 9, 2, 1, 0, 16, 27, 4, 5, 2, 0, 32, 81, 8, 25, 5, 1, 0, 64, 243, 16, 125, 13, 7, 1, 0, 128, 729, 32, 625, 35, 49, 2, 1, 0, 256, 2187, 64, 3125, 97, 343, 4, 3, 2, 0, 512, 6561, 128, 15625, 275, 2401, 8, 9, 7, 1, 0, 1024, 19683, 256, 78125, 793, 16807, 16, 27, 29, 11, 2
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 26 2018

Keywords

Examples

			Square array begins:
  0,  0,   0,    0,    0,     0,  ...
  1,  2,   4,    8,   16,    32,  ...
  1,  3,   9,   27,   81,   243,  ...
  1,  2,   4,    8,   16,    32,  ...
  1,  5,  25,  125,  625,  3125,  ...
  2,  5,  13,   35,   97,   275,  ...

Links

Crossrefs

Columns k=0..4 give A001221, A008472, A005063, A005064, A005065.
Cf. A109974, A200768 (diagonal), A285425, A286880, A321258.

Programs

  • Mathematica
    Table[Function[k, Sum[Boole[PrimeQ[d]] d^k, {d, Divisors[n]}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[Prime[j]^k x^Prime[j]/(1 - x^Prime[j]), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
  • PARI
    T(n,k)={vecsum([p^k | p<-factor(n)[,1]])}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018

Formula

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

A321263 a(n) = [x^n] 1/(1 - Sum_{k>=1} k^n*x^(2*k)/(1 - x^k)).

Original entry on oeis.org

1, 0, 1, 1, 18, 3, 926, 264, 146255, 64190, 138356840, 22816773, 509079790798, 108923489863, 6757117812676818, 1403337110700033, 474610323092906351464, 52144014892723916074, 130074987349483695192896881, 14487112805054799566652854, 132992779975091800967037313578152
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 01 2018

Keywords

Crossrefs

Cf. A321190, A321258, A321260, A321261, A321262.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1 - Sum[k^n x^(2 k)/(1 - x^k), {k, 1, n}]), {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[1/(1 - Sum[(DivisorSigma[n, k] - k^n) x^k, {k, 1, n}]), {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[1/(1 - Sum[(k^n - Sum[d^n MoebiusMu[k/d], {d, Divisors[k]}]) x^k/(1 - x^k), {k, 1, n}]), {x, 0, n}], {n, 0, 20}]

Formula

a(n) = [x^n] 1/(1 - Sum_{k>=1} (sigma_n(k) - k^n)*x^k).
a(n) = [x^n] 1/(1 - Sum_{k>=1} (k^n - J_n(k))*x^k/(1 - x^k)), where J_() is the Jordan function.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.