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.

A350106 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} j * floor(n/j)^k.

Original entry on oeis.org

1, 1, 4, 1, 6, 8, 1, 10, 14, 15, 1, 18, 32, 31, 21, 1, 34, 86, 87, 45, 33, 1, 66, 248, 295, 153, 81, 41, 1, 130, 734, 1095, 669, 309, 101, 56, 1, 258, 2192, 4231, 3201, 1521, 443, 150, 69, 1, 514, 6566, 16647, 15765, 8373, 2633, 722, 191, 87, 1, 1026, 19688, 66055, 78393, 48321, 17411, 4746, 1005, 253, 99
Offset: 1

Views

Author

Seiichi Manyama, Dec 14 2021

Keywords

Examples

			Square array begins:
   1,   1,   1,    1,     1,      1,      1, ...
   4,   6,  10,   18,    34,     66,    130, ...
   8,  14,  32,   86,   248,    734,   2192, ...
  15,  31,  87,  295,  1095,   4231,  16647, ...
  21,  45, 153,  669,  3201,  15765,  78393, ...
  33,  81, 309, 1521,  8373,  48321, 284709, ...
  41, 101, 443, 2633, 17411, 119321, 828323, ...

Links

Crossrefs

Columns k=1..3 give A024916, A350107, A350108.
T(n,n) gives A350109.
Cf. A319649, A344725.

Programs

  • Mathematica
    T[n_, k_] := Sum[j * Floor[n/j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Dec 14 2021 *)
  • PARI
    T(n, k) = sum(j=1, n, j*(n\j)^k);
  • PARI
    T(n, k) = sum(j=1, n, j*sumdiv(j, d, (d^k-(d-1)^k)/d));

Formula

G.f. of column k: (1/(1 - x)) * Sum_{j>=1} (j^k - (j - 1)^k) * x^j/(1 - x^j)^2.
T(n,k) = Sum_{j=1..n} j * Sum_{d|j} (d^k - (d - 1)^k)/d.

A356124 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} j^k * binomial(floor(n/j)+1,2).

Original entry on oeis.org

1, 1, 4, 1, 5, 8, 1, 7, 11, 15, 1, 11, 19, 23, 21, 1, 19, 41, 47, 33, 33, 1, 35, 103, 125, 77, 57, 41, 1, 67, 281, 395, 255, 149, 71, 56, 1, 131, 799, 1373, 1025, 555, 205, 103, 69, 1, 259, 2321, 5027, 4503, 2537, 905, 325, 130, 87, 1, 515, 6823, 18965, 20657, 12867, 4945, 1585, 442, 170, 99
Offset: 1

Views

Author

Seiichi Manyama, Jul 27 2022

Keywords

Examples

			Square array begins:
   1,  1,   1,   1,    1,     1,     1, ...
   4,  5,   7,  11,   19,    35,    67, ...
   8, 11,  19,  41,  103,   281,   799, ...
  15, 23,  47, 125,  395,  1373,  5027, ...
  21, 33,  77, 255, 1025,  4503, 20657, ...
  33, 57, 149, 555, 2537, 12867, 68969, ...

Crossrefs

Column k=0..4 give A024916, A143127, A143128, A356125, A356126.
T(n,n) gives A356129.
T(n,n+1) gives A356128.
Cf. A279394, A319649, A350106.

Programs

  • Mathematica
    T[n_, k_] := Sum[j^k * Binomial[Floor[n/j] + 1, 2], {j, 1, n}]; Table[T[k, n - k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Jul 28 2022 *)
  • PARI
    T(n, k) = sum(j=1, n, j^k*binomial(n\j+1, 2));
  • PARI
    T(n, k) = sum(j=1, n, j*sigma(j, k-1));
  • Python
    from itertools import count, islice
from math import isqrt
from sympy import bernoulli
def A356124_T(n,k): return ((s:=isqrt(n))*(s+1)*(bernoulli(k+1)-bernoulli(k+1,s+1))+sum(w**k*(k+1)*((q:=n//w)*(q+1))+(w*(bernoulli(k+1,q+1)-bernoulli(k+1))<<1) for w in range(1,s+1)))//(k+1)>>1
def A356124_gen(): # generator of terms
     return (A356124_T(k+1,n-k-1) for n in count(1) for k in range(n))
A356124_list = list(islice(A356124_gen(),30)) # Chai Wah Wu, Oct 24 2023

Formula

G.f. of column k: (1/(1-x)) * Sum_{j>=1} j^k * x^j/(1 - x^j)^2.
T(n,k) = Sum_{j=1..n} j * sigma_{k-1}(j).

A308313 a(n) = Sum_{k=1..n} (-1)^(n-k) * k^n * floor(n/k).

Original entry on oeis.org

1, 2, 22, 203, 2285, 33855, 609345, 12420372, 284964519, 7347342215, 209807114169, 6554034238459, 222469737401739, 8159109186320903, 321461264348047819, 13538455640979049698, 606976994365011212414, 28864017965496692865925, 1451086990386146504580735, 76896033641977171208887465
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 22 2019

Keywords

Crossrefs

Cf. A319194, A319649.

Programs

  • Mathematica
    Table[Sum[(-1)^(n - k) k^n Floor[n/k] , {k, 1, n}], {n, 1, 20}]
Table[SeriesCoefficient[1/(1 + x) Sum[k^n x^k/(1 - (-x)^k), {k, 1, n}], {x, 0, n}], {n, 1, 20}]
Table[(-1)^n Sum[DivisorSigma[n, k] - 2 Total[Select[Divisors[k], OddQ]^n], {k, 1, n}], {n, 1, 20}]
  • PARI
    a(n)={sum(k=1, n, (-1)^(n-k) * k^n * (n\k))} \\ Andrew Howroyd, Aug 22 2019
  • Python
    from math import isqrt
from sympy import bernoulli
def A308313(n): return (-1 if n&1 else 1)*((((s:=isqrt(m:=n>>1))+1)*(bernoulli(n+1)-bernoulli(n+1,s+1))<

Formula

a(n) = [x^n] (1/(1 + x)) * Sum_{k>=1} k^n * x^k/(1 - (-x)^k).
a(n) = Sum_{k=1..n} Sum_{d|k} (-1)^(n-d) * d^n.
a(n) ~ c * n^n, where c = 1/(1 + exp(-1)) = 0.7310585786300048792511592418218362743651446401650565192763659... - Vaclav Kotesovec, Aug 22 2019, updated Jul 19 2021
Let A(n,k) = Sum_{j=1..n} j^k * floor(n/j). Then a(n) = (-1)^n*(2^(n+1)*A(floor(n/2),n)-A(n,n)). - Chai Wah Wu, Oct 28 2023

A356130 a(n) = Sum_{k=1..n} sigma_{n-1}(k).

Original entry on oeis.org

1, 4, 16, 111, 999, 12513, 185683, 3316418, 67810767, 1576561677, 40862702931, 1171104916405, 36722498575799, 1251419967587955, 46034784688102781, 1818440444592581068, 76763036794222996512, 3448830049286378614987, 164309958491233496689189
Offset: 1

Views

Author

Seiichi Manyama, Jul 27 2022

Keywords

Links

Crossrefs

Cf. A319194, A319649, A356129.

Programs

  • Mathematica
    a[n_] := Sum[DivisorSigma[n-1, k], {k, 1, n}]; Array[a, 19] (* Amiram Eldar, Jul 28 2022 *)
  • PARI
    a(n) = sum(k=1, n, sigma(k, n-1));
  • PARI
    a(n) = sum(k=1, n, k^(n-1)*(n\k));
  • Python
    from math import isqrt
from sympy import bernoulli
def A350130(n): return (((s:=isqrt(n))+1)*((b:=bernoulli(n))-bernoulli(n, s+1))+sum(k**(n-1)*n*((q:=n//k)+1)-b+bernoulli(n, q+1) for k in range(1,s+1)))//n if n>1 else 1 # Chai Wah Wu, Oct 21 2023

Formula

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

A366919 a(n) = Sum_{k=1..n} (-1)^k*k^n*floor(n/k).

Original entry on oeis.org

-1, 2, -22, 203, -2285, 33855, -609345, 12420372, -284964519, 7347342215, -209807114169, 6554034238459, -222469737401739, 8159109186320903, -321461264348047819, 13538455640979049698, -606976994365011212414, 28864017965496692865925, -1451086990386146504580735
Offset: 1

Views

Author

Chai Wah Wu, Oct 28 2023

Keywords

Crossrefs

Cf. A024919, A308313, A366915, A366917, A319649.

Programs

  • Mathematica
    a[n_]:=Sum[ (-1)^k*k^n*Floor[n/k],{k,n}]; Array[a,19] (* Stefano Spezia, Oct 29 2023 *)
  • PARI
    a(n) = sum(k=1, n, (-1)^k*k^n*(n\k)); \\ Michel Marcus, Oct 29 2023
  • Python
    from math import isqrt
from sympy import bernoulli
def A366919(n): return ((((s:=isqrt(m:=n>>1))+1)*(bernoulli(n+1)-bernoulli(n+1,s+1))<

Formula

a(n) = (-1)^n*A308313(n).
Let A(n,k) = Sum_{j=1..n} j^k * floor(n/j). Then a(n) = 2^(n+1)*A(floor(n/2),n)-A(n,n).

A366936 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (-1)^j*j^k*floor(n/j).

Original entry on oeis.org

-1, -1, -1, -1, 0, -3, -1, 2, -4, -2, -1, 6, -8, 1, -4, -1, 14, -22, 11, -5, -4, -1, 30, -68, 49, -15, -1, -6, -1, 62, -214, 203, -77, 15, -9, -4, -1, 126, -668, 841, -423, 119, -35, 4, -7, -1, 254, -2062, 3491, -2285, 807, -225, 48, -9, -7, -1, 510, -6308, 14449
Offset: 1

Views

Author

Chai Wah Wu, Oct 29 2023

Keywords

Examples

			Array begins:
-1, -1,  -1,  -1,   -1,    -1,     -1,     -1,      -1,       -1, ...
-1,  0,   2,   6,   14,    30,     62,    126,     254,      510, ...
-3, -4,  -8, -22,  -68,  -214,   -668,  -2062,   -6308,   -19174, ...
-2,  1,  11,  49,  203,   841,   3491,  14449,   59483,   243481, ...
-4, -5, -15, -77, -423, -2285, -12135, -63677, -331143, -1709645, ...

Crossrefs

First column is -A059851.
Second column is A024919.
Third column is A366915.
Fourth column is A366917.
First row is -A000012.
Second row is A000918.
First superdiagonal is A366919.
Cf. A319649.

Programs

  • Python
    from math import isqrt
from itertools import count, islice
from sympy import bernoulli
def A366936_T(n,k):
    if k:
        return ((((s:=isqrt(m:=n>>1))+1)*(bernoulli(k+1)-bernoulli(k+1,s+1))<>1))**2<<1)+((sum(m//k for k in range(1, t+1))<<1)-sum(n//k for k in range(1, s+1))<<1)
def A366936_gen(): return (A366936_T(k+1,n-k-1) for n in count(1) for k in range(n))
A366936_list = list(islice(A366936_gen(),30))

Formula

Let A(n, k) = Sum_{j=1..n} j^k * floor(n/j). Then T(n, k) = 2^(k+1)*A(floor(n/2), k) - A(n, k).
Showing 1-6 of 6 results.

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