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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A308502 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} d^(n/d + k).

Original entry on oeis.org

1, 1, 3, 1, 5, 4, 1, 9, 10, 9, 1, 17, 28, 25, 6, 1, 33, 82, 81, 26, 24, 1, 65, 244, 289, 126, 80, 8, 1, 129, 730, 1089, 626, 330, 50, 41, 1, 257, 2188, 4225, 3126, 1604, 344, 161, 37, 1, 513, 6562, 16641, 15626, 8634, 2402, 833, 163, 68, 1, 1025, 19684, 66049, 78126, 49100, 16808, 5249, 973, 290, 12
Offset: 1

Views

Author

Seiichi Manyama, Jun 02 2019

Keywords

Examples

			Square array begins:
    1,  1,   1,    1,    1,     1, ...
    3,  5,   9,   17,   33,    65, ...
    4, 10,  28,   82,  244,   730, ...
    9, 25,  81,  289, 1089,  4225, ...
    6, 26, 126,  626, 3126, 15626, ...
   24, 80, 330, 1604, 8634, 49100, ...

Links

Crossrefs

Columns k=0..2 give A055225, A078308, A296601.
Cf. A279394, A308504.

Programs

  • Mathematica
    T[n_, k_] := DivisorSum[n, #^(n/# + k) &]; Table[T[k, n - k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, May 11 2021 *)

Formula

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

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)).

A308690 Square array A(n,k), n >= 1, k >= 0, where A(n,k) = Sum_{d|n} d^(k*n/d - k + 1), read by antidiagonals.

Original entry on oeis.org

1, 1, 3, 1, 3, 4, 1, 3, 4, 7, 1, 3, 4, 9, 6, 1, 3, 4, 13, 6, 12, 1, 3, 4, 21, 6, 24, 8, 1, 3, 4, 37, 6, 66, 8, 15, 1, 3, 4, 69, 6, 216, 8, 41, 13, 1, 3, 4, 133, 6, 762, 8, 201, 37, 18, 1, 3, 4, 261, 6, 2784, 8, 1289, 253, 68, 12, 1, 3, 4, 517, 6, 10386, 8, 9225, 2197, 648, 12, 28
Offset: 1

Views

Author

Seiichi Manyama, Jun 17 2019

Keywords

Examples

			Square array begins:
    1,  1,  1,   1,   1,    1,     1, ...
    3,  3,  3,   3,   3,    3,     3, ...
    4,  4,  4,   4,   4,    4,     4, ...
    7,  9, 13,  21,  37,   69,   133, ...
    6,  6,  6,   6,   6,    6,     6, ...
   12, 24, 66, 216, 762, 2784, 10386, ...
    8,  8,  8,   8,   8,    8,     8, ...

Links

Crossrefs

Columns k=0..3 give A000203, A055225, A308688, A308689.
Cf. A294579, A308509, A308694.

Programs

  • Mathematica
    T[n_, k_] := DivisorSum[n, #^(k*n/# - k + 1) &]; Table[T[k, n - k], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, May 09 2021 *)

Formula

L.g.f. of column k: -log(Product_{j>=1} (1 - j^k*x^j)^(1/j^k)).
A(p,k) = p+1 for prime p.

A359103 a(n) = Sum_{d|n} d * (n/d)^d.

Original entry on oeis.org

1, 4, 6, 16, 10, 54, 14, 112, 99, 230, 22, 996, 26, 1022, 1620, 3232, 34, 9828, 38, 18100, 16380, 22814, 46, 133272, 15675, 106886, 179388, 354116, 58, 1218150, 62, 1589824, 1952676, 2228870, 630980, 13767264, 74, 9962270, 20732868, 34787000, 82, 113676402, 86
Offset: 1

Views

Author

Seiichi Manyama, Dec 17 2022

Keywords

Links

Crossrefs

Cf. A055225, A087909, A167531, A359112, A359863.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^#*# &]; Array[a, 43] (* Amiram Eldar, Aug 27 2023 *)
  • PARI
    a(n) = sumdiv(n, d, d*(n/d)^d);
  • PARI
    my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, k*x^k/(1-k*x^k)^2))

Formula

a(n) = n * A087909(n).
G.f.: Sum_{k>=1} k * x^k/(1 - k * x^k)^2.
If p is prime, a(p) = 2 * p.
a(n) = [x^n] Sum_{k>0} k * (n * x / k)^k / (1 - x^k). - Seiichi Manyama, Jan 16 2023

A359700 a(n) = Sum_{d|n} d^(d + n/d - 1).

Original entry on oeis.org

1, 5, 28, 265, 3126, 46754, 823544, 16778273, 387420733, 10000015690, 285311670612, 8916100733146, 302875106592254, 11112006831323074, 437893890380939688, 18446744073843786241, 827240261886336764178, 39346408075300026047027
Offset: 1

Views

Author

Seiichi Manyama, Jan 11 2023

Keywords

Links

Crossrefs

Cf. A014566, A055225, A087909, A294956, A353013, A353014.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(# + n/# - 1) &]; Array[a, 20] (* Amiram Eldar, Aug 14 2023 *)
  • PARI
    a(n) = sumdiv(n, d, d^(d+n/d-1));
  • 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>0} (k * x)^k / (1 - k * x^k).
If p is prime, a(p) = 1 + p^p.

A363661 a(n) = Sum_{d|n} (n/d)^d * binomial(d+n,n).

Original entry on oeis.org

2, 12, 32, 150, 282, 1890, 3488, 21582, 54650, 282612, 705564, 4072224, 10400782, 55006530, 158987232, 790611350, 2333606526, 11573213196, 35345264180, 168673694070, 540848064614, 2500462200182, 8233430728152, 37445946291600, 126411051769652
Offset: 1

Views

Author

Seiichi Manyama, Jun 14 2023

Keywords

Comments

All terms are even. - Robert Israel, Nov 23 2023

Links

Crossrefs

Cf. A362683, A363639, A363640.
Cf. A055225, A363660, A363662, A363663.

Programs

  • Maple
    f:= proc(n) local d;
   add((n/d)^d * binomial(n+d,n), d = numtheory:-divisors(n))
end proc:
map(f, [$1..30]); # Robert Israel, Nov 23 2023
  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^# * Binomial[# + n, n] &]; Array[a, 30] (* Amiram Eldar, Jul 05 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^d*binomial(d+n, n));

Formula

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

A073706 a(n) = Sum_{ d divides n } (n/d)^(3d).

Original entry on oeis.org

1, 9, 28, 129, 126, 1458, 344, 8705, 20413, 49394, 1332, 1104114, 2198, 2217546, 16305408, 33820673, 4914, 532253187, 6860, 2392632274, 10500716072, 8591716802, 12168, 422182489826, 30517593751, 549760658274, 7625984925160
Offset: 1

Views

Author

Paul D. Hanna, Aug 04 2002

Keywords

Examples

			a(10) = (10/1)^(3*1) +(10/2)^(3*2) +(10/5)^(3*5) +(10/10)^(3*10) = 49394 because positive divisors of 10 are 1, 2, 5, 10.

Links

Crossrefs

Sum_{ d divides n } (n/d)^(k*d): A000005 (k=0), A055225 (k=1), A073705 (k=2), this sequence (k=3).

Programs

  • Mathematica
    Table[Total[Quotient[n, x = Divisors[n]]^(3*x)], {n, 27}] (* Jayanta Basu, Jul 08 2013 *)

Formula

G.f.: Sum_{n>=1} -log(1 - (n^3)*x^n)/n = Sum_{n>=1} a(n) x^n/n.
G.f.: Sum_{k>=1} k^3*x^k/(1-k^3*x^k). - Benoit Cloitre, Apr 21 2003

A333823 a(n) = Sum_{d|n, d odd} (n/d)^d.

Original entry on oeis.org

1, 2, 4, 4, 6, 14, 8, 8, 37, 42, 12, 76, 14, 142, 384, 16, 18, 746, 20, 1044, 2552, 2070, 24, 536, 3151, 8218, 20440, 16412, 30, 41574, 32, 32, 178512, 131106, 94968, 263908, 38, 524326, 1596560, 32808, 42, 2379874, 44, 4194348, 16364502, 8388654, 48, 4144, 823593, 33654482
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 06 2020

Keywords

Links

Crossrefs

Cf. A055225, A333824.

Programs

  • Mathematica
    Table[DivisorSum[n, (n/#)^# &, OddQ[#] &], {n, 50}]
nmax = 50; CoefficientList[Series[Sum[k x^k/(1 - k^2 x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    a(n) = sumdiv(n, d, if ((d)%2, (n/d)^d)); \\ Michel Marcus, Apr 07 2020
  • Python
    from sympy import divisors
def A333823(n): return sum((n//d)**d for d in divisors(n>>(~n & n-1).bit_length(),generator=True)) # Chai Wah Wu, Jul 09 2023

Formula

G.f.: Sum_{k>=1} k * x^k / (1 - k^2*x^(2*k)).
a(2^n) = 2^n. - Seiichi Manyama, Apr 07 2020

A333824 a(n) = Sum_{d|n, n/d odd} (n/d)^d.

Original entry on oeis.org

1, 1, 4, 1, 6, 10, 8, 1, 37, 26, 12, 82, 14, 50, 384, 1, 18, 811, 20, 626, 2552, 122, 24, 6562, 3151, 170, 20440, 2402, 30, 74900, 32, 1, 178512, 290, 94968, 538003, 38, 362, 1596560, 390626, 42, 4901060, 44, 14642, 16364502, 530, 48, 43046722, 823593, 9766251
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 06 2020

Keywords

Links

Crossrefs

Cf. A055225, A333823.

Programs

  • Mathematica
    Table[DivisorSum[n, (n/#)^# &, OddQ[n/#] &], {n, 50}]
nmax = 50; CoefficientList[Series[Sum[(2 k - 1) x^(2 k - 1)/(1 - (2 k - 1) x^(2 k - 1)), {k, 1,nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    a(n) = sumdiv(n, d, if ((n/d)%2, (n/d)^d)); \\ Michel Marcus, Apr 07 2020
  • Python
    from sympy import divisors
def A333824(n): return sum(d**(n//d) for d in divisors(n>>(~n & n-1).bit_length(),generator=True)) # Chai Wah Wu, Jul 09 2023

Formula

G.f.: Sum_{k>=1} (2*k - 1) * x^(2*k - 1) / (1 - (2*k - 1)*x^(2*k - 1)).
a(2^n) = 1. - Seiichi Manyama, Apr 07 2020
a(n) = Sum_{d|n, d odd} d^(n/d). - Chai Wah Wu, Jul 09 2023

A345100 a(n) = Sum_{k=1..n} k^floor(n/k).

Original entry on oeis.org

1, 3, 6, 12, 17, 33, 40, 68, 95, 141, 152, 328, 341, 461, 738, 1130, 1147, 2159, 2178, 4068, 5841, 6997, 7020, 18198, 20723, 25001, 38798, 61546, 61575, 137445, 137476, 223252, 342593, 408435, 485376, 1213988, 1214025, 1476549, 2541498, 4202810, 4202851, 8777205
Offset: 1

Views

Author

Seiichi Manyama, Jun 08 2021

Keywords

Links

Crossrefs

Cf. A031971, A055225, A344551, A345098.

Programs

  • Mathematica
    a[n_] := Sum[k^Floor[n/k], {k, 1, n}]; Array[a, 40] (* Amiram Eldar, Jun 08 2021 *)
  • PARI
    a(n) = sum(k=1, n, k^(n\k));
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, k*x^k*(1-x^k)/(1-k*x^k))/(1-x))

Formula

G.f.: (1/(1 - x)) * Sum_{k>=1} k*x^k * (1 - x^k)/(1 - k*x^k).
a(n) ~ 3^((n - mod(n,3))/3). - Vaclav Kotesovec, Jun 11 2021
Previous Showing 21-30 of 45 results. Next

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