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 1-6 of 6 results.

A332533 a(n) = (1/n) * Sum_{k=1..n} floor(n/k) * n^k.

Original entry on oeis.org

1, 4, 15, 92, 790, 9384, 137326, 2397352, 48428487, 1111122360, 28531183329, 810554859732, 25239592620853, 854769763924104, 31278135039463245, 1229782938533709200, 51702516368332126932, 2314494592676172411516, 109912203092257573556274, 5518821052632117898282620
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 16 2020

Keywords

Links

Crossrefs

Cf. A024916, A268235, A308313, A308814, A319194, A320095.
Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), A344818 (q=-3), A344817 (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), this sequence (q=n).

Programs

  • Magma
    A332533:= func< n | (&+[Floor(n/j)*n^(j-1): j in [1..n]]) >;
[A332533(n): n in [1..40]]; // G. C. Greubel, Jun 27 2024
  • Maple
    seq(add(n^(k-1)*floor(n/k), k=1..n), n=1..60); # Ridouane Oudra, Mar 05 2023
  • Mathematica
    Table[(1/n) Sum[Floor[n/k] n^k, {k, 1, n}], {n, 1, 20}]
Table[(1/n) Sum[Sum[n^d, {d, Divisors[k]}], {k, 1, n}], {n, 1, 20}]
Table[SeriesCoefficient[(1/(1 - x)) Sum[x^k/(1 - n x^k), {k, 1, n}], {x, 0, n}], {n, 1, 20}]
  • PARI
    a(n) = sum(k=1, n, (n\k)*n^k)/n; \\ Michel Marcus, Feb 16 2020
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, n^(d-1))); \\ Seiichi Manyama, May 29 2021
  • SageMath
    def A332533(n): return sum((n//j)*n^(j-1) for j in range(1,n+1))
[A332533(n) for n in range(1,41)] # G. C. Greubel, Jun 27 2024

Formula

a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} x^k / (1 - n*x^k).
a(n) = (1/n) * Sum_{k=1..n} Sum_{d|k} n^d.
a(n) ~ n^(n-1). - Vaclav Kotesovec, May 28 2021
a(n) = (1/(n-1)) * Sum_{k=1..n} (n^floor(n/k) - 1), for n>=2. - Ridouane Oudra, Mar 05 2023

A332508 a(n) = Sum_{d|n} binomial(n+d-2, n-1).

Original entry on oeis.org

1, 3, 7, 25, 71, 280, 925, 3561, 12916, 49346, 184757, 710255, 2704157, 10427747, 40119781, 155288897, 601080391, 2334714319, 9075135301, 35352181116, 137846759282, 538302226628, 2104098963721, 8233718962365, 32247603703576, 126412458920775, 495918551104687
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 14 2020

Keywords

Links

Crossrefs

Cf. A000005, A000203, A007437, A059358, A073570, A101289, A308814, A332470.

Programs

  • Mathematica
    Table[DivisorSum[n, Binomial[n + # - 2, n - 1] &], {n, 1, 27}]
Table[SeriesCoefficient[Sum[x^k/(1 - x^k)^n, {k, 1, n}], {x, 0, n}], {n, 1, 27}]
  • PARI
    a(n) = sumdiv(n, d, binomial(n+d-2, n-1)); \\ Michel Marcus, Feb 14 2020

Formula

a(n) = [x^n] Sum_{k>=1} x^k / (1 - x^k)^n.
a(n) ~ 4^(n-1) / sqrt(Pi*n). - Vaclav Kotesovec, Aug 04 2022

A308813 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where A(n,k) is Sum_{d|n} k^(d-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 5, 3, 1, 1, 5, 10, 11, 2, 1, 1, 6, 17, 31, 17, 4, 1, 1, 7, 26, 69, 82, 39, 2, 1, 1, 8, 37, 131, 257, 256, 65, 4, 1, 1, 9, 50, 223, 626, 1045, 730, 139, 3, 1, 1, 10, 65, 351, 1297, 3156, 4097, 2218, 261, 4, 1
Offset: 1

Views

Author

Seiichi Manyama, Jun 26 2019

Keywords

Examples

			Square array, A(n,k), begins:
  1, 1,  1,   1,    1,     1,     1, ...
  1, 2,  3,   4,    5,     6,     7, ...
  1, 2,  5,  10,   17,    26,    37, ...
  1, 3, 11,  31,   69,   131,   223, ...
  1, 2, 17,  82,  257,   626,  1297, ...
  1, 4, 39, 256, 1045,  3156,  7819, ...
  1, 2, 65, 730, 4097, 15626, 46657, ...
Antidiagonal triangle, T(n,k), begins as:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  2,   1;
  1,  4,  5,   3,    1;
  1,  5, 10,  11,    2,    1;
  1,  6, 17,  31,   17,    4,    1;
  1,  7, 26,  69,   82,   39,    2,    1;
  1,  8, 37, 131,  257,  256,   65,    4,   1;
  1,  9, 50, 223,  626, 1045,  730,  139,   3,   1;
  1, 10, 65, 351, 1297, 3156, 4097, 2218, 261,   4,   1;

Links

Crossrefs

Columns k=0..10 give A000012, A000005, A034729, A034730, A339684, A339685, A339686, A339687, A339688, A339689, A113999.
Row n=1..3 give A000012, A000027(k+1), A002522.
A(n,n) gives A308814.

Programs

  • Magma
    A:= func< n,k | (&+[k^(d-1): d in Divisors(n)]) >;
A308813:= func< n,k | A(k+1,n-k-1) >;
[A308813(n,k): k in [0..n-1], n in [1..12]]; // G. C. Greubel, Jun 26 2024
  • Mathematica
    A[n_, k_] := DivisorSum[n, If[k == # - 1 == 0, 1, k^(# - 1)] &];
Table[A[k + 1, n - k - 1], {n, 1, 11}, {k, 0, n - 1}] // Flatten (* Amiram Eldar, May 07 2021 *)
  • SageMath
    def A(n,k): return sum(k^(j-1) for j in (1..n) if (j).divides(n))
def A308813(n,k): return A(k+1,n-k-1)
flatten([[A308813(n,k) for k in range(n)] for n in range(1,13)]) # G. C. Greubel, Jun 26 2024

Formula

G.f. of column k: Sum_{j>=1} x^j/(1 - k*x^j).
T(n, k) = Sum_{d|(k+1)} (n-k-1)^(d-1), with T(n, n) = 1. - G. C. Greubel, Jun 26 2024

A332621 a(n) = (1/n) * Sum_{k=1..n} n^(n/gcd(n, k)).

Original entry on oeis.org

1, 3, 19, 133, 2501, 15631, 705895, 8389641, 258280489, 4000040011, 259374246011, 2972033984173, 279577021469773, 4762288684702095, 233543408203327951, 9223372037928525841, 778579070010669895697, 13115469358498302735067, 1874292305362402347591139
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 17 2020

Keywords

Links

Crossrefs

Cf. A000010, A056665, A130586, A226561, A228640, A308814, A321349, A332620.

Programs

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

Formula

a(n) = [x^n] Sum_{k>=1} Sum_{j>=1} phi(j) * n^(j-1) * x^(k*j).
a(n) = (1/n) * Sum_{k=1..n} n^(lcm(n, k)/k).
a(n) = (1/n) * Sum_{d|n} phi(d) * n^d.
a(n) = A332620(n) / n.

A383003 a(n) = Sum_{d|n} (-n)^(d-1).

Original entry on oeis.org

1, -1, 10, -67, 626, -7745, 117650, -2097671, 43046803, -999990009, 25937424602, -743008621115, 23298085122482, -793714765724621, 29192926025441476, -1152921504875286543, 48661191875666868482, -2185911559727678460653, 104127350297911241532842
Offset: 1

Views

Author

Seiichi Manyama, Apr 12 2025

Keywords

Crossrefs

Cf. A101561, A101562, A101563.
Cf. A262843, A308814, A383010.
Cf. A382998, A383012.

Programs

  • PARI
    a(n) = sumdiv(n, d, (-n)^(d-1));

Formula

a(n) = (1/n) * A383010(n).
a(n) = [x^n] Sum_{k>=1} log(1 + n*x^k) / k.
a(n) = [x^n] Sum_{k>=1} x^k / (1 + n*x^k).

A332411 If n = Product (p_j^k_j) then a(n) = Sum (n^(pi(p_j) - 1)), where pi = A000720.

Original entry on oeis.org

0, 1, 3, 1, 25, 7, 343, 1, 9, 101, 14641, 13, 371293, 2745, 240, 1, 24137569, 19, 893871739, 401, 9282, 234257, 78310985281, 25, 625, 11881377, 27, 21953, 14507145975869, 931, 819628286980801, 1, 1185954, 1544804417, 44100, 37, 177917621779460413, 114415582593, 90224238, 1601
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 11 2020

Keywords

Examples

			a(21) = a(3 * 7) = a(prime(2) * prime(4)) = 21^1 + 21^3 = 9282;
9282 in base 21 (reverse order of digits with leading zero) = 0101.
                                                               | |
                                                               2 4

Links

Crossrefs

Cf. A000079 (without a(0) gives the positions of 1's), A000244 (without a(0) gives the fixed points), A000720, A087207, A090883, A276379 (a(n) written in base n), A308814.

Programs

  • Maple
    a:= n-> add(n^(numtheory[pi](i[1])-1), i=ifactors(n)[2]):
seq(a(n), n=1..42);  # Alois P. Heinz, Feb 11 2020
  • Mathematica
    a[n_] := Plus @@ (n^(PrimePi[#[[1]]] - 1) & /@ FactorInteger[n]); a[1] = 0; Table[a[n], {n, 1, 40}]
Table[SeriesCoefficient[Sum[n^(k - 1) x^Prime[k]/(1 - x^Prime[k]), {k, 1, n}], {x, 0, n}], {n, 1, 40}]
  • PARI
    a(n) = my(f=factor(n)); sum(k=1, #f~, n^(primepi(f[k,1])-1)); \\ Michel Marcus, Feb 11 2020

Formula

a(n) = [x^n] Sum_{k>=1} n^(k - 1) * x^prime(k) / (1 - x^prime(k)).
Showing 1-6 of 6 results.

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