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.

A332623 a(n) = Sum_{k=1..n} ceiling(n/k)^2.

Original entry on oeis.org

1, 5, 14, 25, 43, 58, 87, 106, 141, 171, 212, 239, 302, 333, 386, 439, 507, 546, 631, 674, 765, 834, 911, 962, 1091, 1157, 1246, 1331, 1450, 1513, 1666, 1733, 1866, 1967, 2080, 2181, 2373, 2452, 2577, 2694, 2883, 2970, 3171, 3262, 3437, 3600, 3749, 3848, 4107, 4225
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 17 2020

Keywords

Crossrefs

Cf. A000005, A000203, A006218, A006590, A024916, A222548, A332490, A332569, A332624.

Programs

  • Magma
    [&+[Ceiling(n/k)^2:k in [1..n]]:n in [1..50]]; // Marius A. Burtea, Feb 17 2020
  • Mathematica
    Table[Sum[Ceiling[n/k]^2, {k, 1, n}], {n, 1, 50}]
Table[n + Sum[2 DivisorSigma[1, k] + DivisorSigma[0, k], {k, 1, n - 1}], {n, 1, 50}]
nmax = 50; CoefficientList[Series[x/(1 - x)^2 + x/(1 - x) Sum[(2 k + 1) x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • Python
    from math import isqrt
def A332623(n): return n-(s:=isqrt(n-1))**2*(s+2)+sum((q:=(n-1)//k)*((k<<1)+q+3) for k in range(1,s+1)) # Chai Wah Wu, Oct 24 2023

Formula

G.f.: x/(1 - x)^2 + (x/(1 - x)) * Sum_{k>=1} (2*k + 1) * x^k / (1 - x^k).
a(n) = n + Sum_{k=1..n-1} (2*sigma(k) + d(k)).
a(n) ~ n^2 * Pi^2 / 6. - Vaclav Kotesovec, Feb 20 2020

A332569 a(n) = Sum_{k=1..n} floor(n/k) * ceiling(n/k).

Original entry on oeis.org

1, 5, 12, 23, 36, 54, 74, 97, 125, 156, 186, 226, 268, 306, 354, 409, 458, 515, 574, 636, 710, 778, 838, 922, 1013, 1086, 1168, 1264, 1350, 1452, 1556, 1651, 1762, 1864, 1966, 2105, 2234, 2332, 2448, 2594, 2726, 2864, 3004, 3132, 3294, 3444, 3564, 3736, 3917, 4067
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 16 2020

Keywords

Crossrefs

Row sums of A085383.
Cf. A000203, A006218, A006590, A024916, A049697, A222548, A332490.

Programs

  • Magma
    [&+[Floor(n/k)*Ceiling(n/k):k in [1..n]]:n in [1..50]]; // Marius A. Burtea, Feb 16 2020
  • Mathematica
    Table[Sum[Ceiling[n/k] Floor[n/k], {k, 1, n}], {n, 1, 50}]
Table[1 + Sum[DivisorSigma[1, k] + DivisorSigma[1, k + 1], {k, 1, n - 1}], {n, 1, 50}]
nmax = 50; CoefficientList[Series[((1 + x)/(1 - x)) Sum[x^k/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    a(n) = sum(k=1, n, my(q=n/k); floor(q) * ceil(q)); \\ Michel Marcus, Feb 17 2020
  • Python
    from math import isqrt
from sympy import divisor_sigma
def A332569(n): return -(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))-divisor_sigma(n) # Chai Wah Wu, Oct 23 2023

Formula

G.f.: ((1 + x) / (1 - x)) * Sum_{k>=1} x^k / (1 - x^k)^2.
a(n) = 1 + Sum_{k=1..n-1} (sigma(k) + sigma(k+1)) for n > 0.
a(n) ~ (Pi*n)^2/6. - Vaclav Kotesovec, Jun 24 2021

A332624 a(n) = Sum_{k=1..n} ceiling(n/k)^n.

Original entry on oeis.org

1, 5, 36, 289, 3433, 47578, 842499, 16850338, 389415029, 10010878371, 285679026506, 8918295095267, 302973286652448, 11112691430262573, 437929106387544254, 18447028378472722051, 827256956775203666857, 39346558275376372606086, 1978429667078835508142129
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 17 2020

Keywords

Crossrefs

Cf. A006590, A031971, A332469, A332490, A332623.

Programs

  • Magma
    [&+[Ceiling(n/k)^n:k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 17 2020
  • Mathematica
    Table[Sum[Ceiling[n/k]^n, {k, 1, n}], {n, 1, 19}]
Table[n + Sum[Sum[(d + 1)^n - d^n, {d, Divisors[k]}], {k, 1, n - 1}], {n, 1, 19}]
Table[SeriesCoefficient[x/(1 - x)^2 + x/(1 - x) Sum[((k + 1)^n - k^n) x^k/(1 - x^k), {k, 1, n}], {x, 0, n}], {n, 1, 19}]

Formula

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

A332846 a(1) = 1; a(n+1) = Sum_{k=1..n} a(k) * ceiling(n/k).

Original entry on oeis.org

1, 1, 3, 8, 20, 50, 121, 297, 716, 1739, 4198, 10157, 24513, 59246, 143006, 345381, 833792, 2013272, 4860337, 11734717, 28329772, 68396030, 165121957, 398644144, 962410246, 2323475153, 5609360573, 13542220814, 32693802921, 78929886033, 190553574988, 460037180829, 1110627936647
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 26 2020

Keywords

Crossrefs

Cf. A006590, A014668, A097919, A332490.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = Sum[a[k] Ceiling[(n - 1)/k], {k, 1, n - 1}]; Table[a[n], {n, 1, 33}]
a[1] = 1; a[n_] := a[n] = a[n - 1] + Sum[a[k] + Sum[a[d], {d, Divisors[k]}], {k, 1, n - 2}]; Table[a[n], {n, 1, 33}]
terms = 33; A[] = 0; Do[A[x] = x (1 + (1/(1 - x)) (A[x] + x Sum[A[x^k], {k, 1, terms}])) + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest

Formula

G.f. A(x) satisfies: A(x) = x * (1 + (1/(1 - x)) * (A(x) + x * Sum_{k>=1} A(x^k))).
a(1) = 1; a(n) = a(n-1) + Sum_{k=1..n-2} (a(k) + Sum_{d|k} a(d)).
a(n) ~ c * (1 + sqrt(2))^n, where c = 0.2594006517235012546870541901936538347053403598092060748627156661727... - Vaclav Kotesovec, Mar 10 2020

A333194 a(n) = Sum_{k=1..n} (ceiling(n/k) mod 2) * k.

Original entry on oeis.org

1, 2, 4, 4, 8, 8, 11, 11, 19, 16, 21, 21, 30, 30, 37, 29, 45, 45, 51, 51, 66, 56, 67, 67, 88, 83, 96, 84, 105, 105, 112, 112, 144, 130, 147, 135, 159, 159, 178, 162, 197, 197, 208, 208, 241, 209, 232, 232, 277, 270, 290, 270, 309, 309, 324, 308, 357, 335, 364, 364
Offset: 1

Views

Author

Ilya Gutkovskiy, May 25 2020

Keywords

Crossrefs

Cf. A000217, A000593, A078471, A120885, A330926, A332490.

Programs

  • Maple
    b:= n-> add(d, d=select(x-> x::odd, numtheory[divisors](n))):
a:= proc(n) option remember; n+`if`(n<2, 0, a(n-1))-b(n-1) end:
seq(a(n), n=1..60);  # Alois P. Heinz, May 25 2020
  • Mathematica
    Table[Sum[Mod[Ceiling[n/k], 2] k, {k, 1, n}], {n, 1, 60}]
Table[n (n + 1)/2 - Sum[DivisorSum[k, (-1)^(k/# + 1) # &], {k, 1, n - 1}], {n, 1, 60}]
nmax = 60; CoefficientList[Series[x/(1 - x) (1/(1 - x)^2 - Sum[k x^k/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x] // Rest
  • PARI
    a(n) = sum(k=1, n, (ceil(n/k) % 2)*k); \\ Michel Marcus, May 26 2020

Formula

G.f.: (x/(1 - x)) * (1/(1 - x)^2 - Sum_{k>=1} k * x^k / (1 + x^k)).
a(n) = n*(n + 1)/2 - Sum_{k=1..n-1} A000593(k).
a(n) = A000217(n) - A078471(n-1).

A333505 a(n) = Sum_{k=1..n} (-1)^(k+1) * k * ceiling(n/k).

Original entry on oeis.org

1, 0, 2, 2, 2, 2, 5, 5, 1, 4, 9, 9, 2, 2, 9, 17, 5, 5, 11, 11, 2, 12, 23, 23, -4, 1, 14, 26, 15, 15, 22, 22, -6, 8, 25, 37, 9, 9, 28, 44, 7, 7, 18, 18, 3, 35, 58, 58, -9, -2, 18, 38, 21, 21, 36, 52, 5, 27, 56, 56, -3, -3, 28, 68, 8, 26, 45, 45, 24, 50, 73, 73, -23, -23, 14
Offset: 1

Views

Author

Ilya Gutkovskiy, May 25 2020

Keywords

Crossrefs

Cf. A001057, A002129, A006590, A024916, A024919, A332490, A332682.

Programs

  • Mathematica
    Table[Sum[(-1)^(k + 1) k Ceiling[n/k], {k, 1, n}], {n, 1, 75}]
Table[(-1)^(n + 1) Ceiling[n/2] + Sum[DivisorSum[k, (-1)^(# + 1) # &], {k, 1, n - 1}], {n, 1, 75}]
nmax = 75; CoefficientList[Series[x/(1 - x) (1/(1 + x)^2 + Sum[(-1)^(k + 1) k x^k/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x] // Rest
  • PARI
    a(n) = sum(k=1, n, (-1)^(k+1)*k*ceil(n/k)); \\ Michel Marcus, May 26 2020
  • Python
    from math import isqrt
def A333505(n): return ((s:=isqrt(m:=n-1>>1))**2*(s+1)-sum((q:=m//k)*((k<<1)+q+1) for k in range(1,s+1))<<1)-((t:=isqrt(n-1))**2*(t+1)-sum((q:=(n-1)//k)*((k<<1)+q+1) for k in range(1,t+1))>>1) + (m+1 if n&1 else -m-1) # Chai Wah Wu, Oct 30 2023

Formula

G.f.: (x/(1 - x)) * (1/(1 + x)^2 + Sum_{k>=1} (-1)^(k+1) * k * x^k / (1 - x^k)).
a(n) = (-1)^(n+1) * ceiling(n/2) + Sum_{k=1..n-1} A002129(k).
a(n) = A001057(n) - A024919(n-1).
Showing 1-6 of 6 results.

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