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 1-8 of 8 results.

A308814 a(n) = Sum_{d|n} n^(d-1).

Original entry on oeis.org

1, 3, 10, 69, 626, 7819, 117650, 2097673, 43046803, 1000010011, 25937424602, 743008621405, 23298085122482, 793714780783695, 29192926025441476, 1152921504875286545, 48661191875666868482, 2185911559749718382455, 104127350297911241532842, 5242880000000512000168021
Offset: 1

Views

Author

Seiichi Manyama, Jun 26 2019

Keywords

Links

Crossrefs

Cf. A066108, A082245, A262843, A308813.

Programs

  • Mathematica
    Table[Total[n^(Divisors[n]-1)],{n,20}] (* Harvey P. Dale, Aug 08 2019 *)
  • PARI
    {a(n) = sumdiv(n, d, n^(d-1))}

Formula

a(n) = A308813(n,n).
a(n) = A066108(n)/n.
a(n) ~ n^(n-1). - Vaclav Kotesovec, Jun 05 2021

A332620 a(n) = Sum_{k=1..n} n^(n/gcd(n, k)).

Original entry on oeis.org

1, 6, 57, 532, 12505, 93786, 4941265, 67117128, 2324524401, 40000400110, 2853116706121, 35664407810076, 3634501279107049, 66672041585829330, 3503151123049919265, 147573952606856413456, 13235844190181388226849, 236078448452969449231206, 35611553801885644604231641
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 17 2020

Keywords

Links

Crossrefs

Cf. A000010, A056665, A066108, A226561, A228640, A321349, A332621.

Programs

  • Magma
    [&+[n^(n div Gcd(n,k)):k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 17 2020
  • Mathematica
    Table[Sum[n^(n/GCD[n, k]), {k, 1, n}], {n, 1, 19}]
Table[Sum[EulerPhi[d] n^d, {d, Divisors[n]}], {n, 1, 19}]
Table[SeriesCoefficient[Sum[Sum[EulerPhi[j] n^j x^(k j), {j, 1, n}], {k, 1, n}], {x, 0, n}], {n, 1, 19}]
  • PARI
    a(n) = sum(k=1, n, n^(n/gcd(n, k))); \\ Michel Marcus, Mar 10 2021

Formula

a(n) = [x^n] Sum_{k>=1} Sum_{j>=1} phi(j) * n^j * x^(k*j).
a(n) = Sum_{k=1..n} n^(lcm(n, k)/k).
a(n) = Sum_{d|n} phi(d) * n^d.
a(n) = n * A332621(n).

A217576 a(n) = Sum_{d divides n} (d!)^(n/d).

Original entry on oeis.org

1, 3, 7, 29, 121, 765, 5041, 40913, 363097, 3643233, 39916801, 479535185, 6227020801, 87203692929, 1307676103777, 20924415922433, 355687428096001, 6402505760917569, 121645100408832001, 2432915176581403649, 51090942299733783937, 1124002321128529922049
Offset: 1

Views

Author

Joerg Arndt, Oct 07 2012

Keywords

Links

Crossrefs

Cf. A062363 ( Sum_{d divides n} d! ).
Cf. A062796 ( Sum_{d divides n} d^d ), A066108 ( Sum_{d divides n} n^d ).

Programs

  • Mathematica
    f[n_]=With[{d=Divisors[n]},Total[(d!)^(n/d)]]; Array[f,25] (* Harvey P. Dale, Dec 20 2023 *)
  • PARI
    a(n)=sumdiv(n,d, (d!)^(n/d) );

Formula

G.f.: Sum_{n>=1} n!*x^n / (1 - n!*x^n). - Paul D. Hanna, Jan 17 2013

A363733 Array read by upwards antidiagonals. The family of polynomials generated by the divisibility matrix (A113704) evaluated over the nonnegative integers.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 6, 3, 1, 0, 3, 10, 12, 4, 1, 0, 2, 22, 30, 20, 5, 1, 0, 4, 34, 93, 68, 30, 6, 1, 0, 2, 78, 246, 276, 130, 42, 7, 1, 0, 4, 130, 768, 1028, 655, 222, 56, 8, 1, 0, 3, 278, 2190, 4180, 3130, 1338, 350, 72, 9, 1
Offset: 0

Views

Author

Peter Luschny, Jun 27 2023

Keywords

Comments

The name expresses the 'row view' of the array. The 'column view' regards the array as the collection of the inverse Möbius transforms of the power sequences k^n = 0^n, 1^n, 2^n, .... (n >= 0). Viewed this way, the array is a generalization of the number of divisors sequence tau (A000005), to which it reduces in the case k = 1.
The array has offset (0, 0). It uses the usual definition of 'k divides n' as described in Apostol, rather than the shortened version, which restricts to values k > 0 as some programs do (but not SageMath). Such a restriction makes sense in the context of rational numbers but not in the case of natural numbers.

Examples

			Array A(n, k) starts:
  [0] 1, 1,   1,    1,     1,      1,       1,       1,        1, ... A000012
  [1] 0, 1,   2,    3,     4,      5,       6,       7,        8, ... A001477
  [2] 0, 2,   6,   12,    20,     30,      42,      56,       72, ... A002378
  [3] 0, 2,  10,   30,    68,    130,     222,     350,      520, ... A034262
  [4] 0, 3,  22,   93,   276,    655,    1338,    2457,     4168, ...
  [5] 0, 2,  34,  246,  1028,   3130,    7782,   16814,    32776, ... A131471
  [6] 0, 4,  78,  768,  4180,  15780,   46914,  118048,   262728, ...
  [7] 0, 2, 130, 2190, 16388,  78130,  279942,  823550,  2097160, ... A190578
  [8] 0, 4, 278, 6654, 65812, 391280, 1680954, 5767258, 16781384, ...
   A000005,A055895,A363913, ...                             A066108 (diagonal)
.
Triangle T(n, k) starts:
  [0] 1;
  [1] 0, 1;
  [2] 0, 1,   1;
  [3] 0, 2,   2,   1;
  [4] 0, 2,   6,   3,    1;
  [5] 0, 3,  10,  12,    4,   1;
  [6] 0, 2,  22,  30,   20,   5,   1;
  [7] 0, 4,  34,  93,   68,  30,   6,  1;
  [8] 0, 2,  78, 246,  276, 130,  42,  7, 1;
  [9] 0, 4, 130, 768, 1028, 655, 222, 56, 8, 1;

References

  • Tom M. Apostol, Introduction to Analytic Number Theory, Springer 1976, p. 14.

Crossrefs

Cf. A113704 (in compact form A113705), A000005 (column 1), A055895 (column 2), A363913 (column 3), A001477 (row 1), A002378 (row 2), A034262 (row 3), A131471 (row 5), A190578 (row 7), A363912 (row sums), A066108 (main diagonal of array).
Cf. A363734, A363735, A363421.

Programs

  • Maple
    divides := (k, n) -> ifelse(k = n or (k > 0 and irem(n, k) = 0), 1, 0):
A := (n, k) -> local j; add(divides(j, n) * k^j, j = 0 ..n):
for n from 0 to 8 do seq(A(n, k), k = 0..8) od;
# If we introduce the 'inverse Möbius transform' InvMoebius acting on s ...
InvMoebius := (s, n) -> local j; add(divides(j, n) * s(j), j = 0 ..n):
# ... the transposed array is given by applying InvMoebius to the powers r^m:
seq(lprint(seq(InvMoebius(m -> r^m, n), n = 0..8)), r = 0..8);
# For instance we see that the number of divisors is the inverse
# Moebius transform of the constant sequence s = 1.
  • SageMath
    def A(n, k): return sum(j.divides(n) * k^j for j in (0..n))
for n in srange(9): print([A(n, k) for k in (0..8)])

Formula

A(n, k) = Sum_{j=0..n} divides(j, n) * k^j, where divides(k, n) <-> [k = n or (k > 0 and n mod k = 0)], and '[ ]' denotes the Iverson bracket.
The columns are the inverse Möbius transforms of the powers x^n, x >= 0.

A383010 a(n) = -Sum_{d|n} (-n)^d.

Original entry on oeis.org

1, -2, 30, -268, 3130, -46470, 823550, -16781368, 387421227, -9999900090, 285311670622, -8916103453380, 302875106592266, -11112006720144694, 437893890381622140, -18446744078004584688, 827240261886336764194, -39346408075098212291754, 1978419655660313589123998
Offset: 1

Views

Author

Seiichi Manyama, Apr 12 2025

Keywords

Crossrefs

Cf. A066108, A383003.

Programs

  • PARI
    a(n) = -sumdiv(n, d, (-n)^d);

A359882 a(n) = Sum_{d|n} d^n * (n/d)^d.

Original entry on oeis.org

1, 6, 30, 324, 3130, 53070, 823550, 17829896, 387951939, 10312525610, 285311670622, 9056807631948, 302875106592266, 11198819379685518, 437901307945957140, 18518802767263301648, 827240261886336764194, 39423330565860716459946
Offset: 1

Views

Author

Seiichi Manyama, Jan 16 2023

Keywords

Crossrefs

Cf. A066108, A308594, A359103.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^n * (n/#)^# &]; Array[a, 20] (* Amiram Eldar, Aug 09 2023 *)
  • PARI
    a(n) = sumdiv(n, d, d^n*(n/d)^d);

Formula

a(n) = [x^n] Sum_{k>0} (n * x)^k / (1 - (k * x)^k).
If p is prime, a(p) = p + p^p.

A336999 a(n) = n! * Sum_{d|n} n^d / d!.

Original entry on oeis.org

1, 8, 45, 544, 3725, 89856, 858823, 25271296, 434776329, 13241728000, 285750755411, 11494661861376, 302956057862653, 12945137688641536, 446924199188379375, 20735627677666902016, 827246308572614396177, 43155924331583693389824
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 10 2020

Keywords

Crossrefs

Cf. A057625, A063170, A066108.

Programs

  • Mathematica
    Table[n! Sum[n^d/d!, {d, Divisors[n]}], {n, 1, 18}]
Table[n! SeriesCoefficient[Sum[(Exp[n x^k] - 1), {k, 1, n}], {x, 0, n}], {n, 1, 18}]
  • PARI
    a(n) = n! * sumdiv(n, d, n^d/d!); \\ Michel Marcus, Aug 12 2020

Formula

a(n) = n! * [x^n] Sum_{k>=1} (exp(n*x^k) - 1).

A348144 a(n) = Sum_{d|n} n^(n^2/d).

Original entry on oeis.org

1, 20, 19710, 4295033088, 298023223876956250, 10314424798490637108305446464, 256923577521058878088611477224235622145150, 6277101735386680763835789423286894578616620063532571951104, 196627050475552913618075908526912116283103450944214825077052418578026469007467
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 02 2021

Keywords

Crossrefs

Cf. A066108, A308593.

Programs

  • Mathematica
    Table[Sum[n^(n^2/i)*(1 - Ceiling[n/i] + Floor[n/i]), {i, n}], {n, 10}]
  • PARI
    a(n) = sumdiv(n, d, n^(n^2/d)); \\ Michel Marcus, Oct 03 2021
Showing 1-8 of 8 results.

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