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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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).

A332681 a(n) = Sum_{k=1..n} mu(k) * ceiling(n/k)^2.

Original entry on oeis.org

1, 3, 4, 8, 11, 20, 23, 35, 43, 56, 63, 83, 90, 115, 128, 144, 159, 191, 202, 238, 255, 280, 299, 343, 359, 400, 424, 460, 483, 538, 553, 613, 646, 687, 720, 768, 791, 864, 901, 949, 980, 1059, 1082, 1166, 1206, 1255, 1298, 1390, 1422, 1506, 1547, 1611, 1658, 1762
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 19 2020

Keywords

Crossrefs

Cf. A000010, A002088, A002321, A008683, A006590, A018805, A206350, A332623.

Programs

  • Mathematica
    Table[Sum[MoebiusMu[k] Ceiling[n/k]^2, {k, 1, n}], {n, 1, 54}]
Join[{1}, Table[2 + Sum[2 EulerPhi[k - 1] + MoebiusMu[k], {k, 2, n}], {n, 2, 54}]]
  • PARI
    a(n) = sum(k=1, n, moebius(k)*ceil(n/k)^2); \\ Michel Marcus, Feb 21 2020

Formula

G.f.: (1/(1 - x)) * (x^2 + Sum_{k>=1} mu(k) * x^k * (1 + 2*x - 2*x^k + x^(2*k)) / (1 - x^k)^2).
a(n) = 2 + Sum_{k=2..n} (2 * phi(k-1) + mu(k)) for n > 1.
a(n) = 1 + 2 * A002088(n-1) + A002321(n) for n > 1.

A332683 a(n) = Sum_{k=1..n, gcd(n, k) = 1} ceiling(n/k).

Original entry on oeis.org

1, 2, 5, 6, 12, 8, 20, 15, 23, 18, 37, 19, 47, 28, 38, 37, 66, 31, 76, 41, 61, 52, 96, 44, 96, 63, 89, 66, 129, 49, 141, 84, 109, 88, 129, 72, 176, 101, 132, 95, 198, 77, 210, 116, 142, 129, 232, 99, 226, 122, 186, 144, 269, 114, 232, 149, 214, 169, 305, 110
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 19 2020

Keywords

Comments

Moebius transform of A006590.

Crossrefs

Cf. A006590, A008683, A116477, A332682.

Programs

  • Mathematica
    Table[Sum[Boole[GCD[n, k] == 1] Ceiling[n/k], {k, 1, n}], {n, 1, 60}]
  • PARI
    a(n) = sum(k=1, n, if (gcd(n, k) == 1, ceil(n/k))); \\ Michel Marcus, Feb 21 2020

Formula

a(n) = Sum_{d|n} mu(n/d) * A006590(d).

A332687 a(n) = Sum_{k=1..n} ceiling(n/prime(k)).

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 13, 15, 17, 19, 22, 24, 27, 29, 32, 35, 37, 39, 42, 44, 47, 50, 53, 55, 58, 60, 63, 65, 68, 70, 74, 76, 78, 81, 84, 87, 90, 92, 95, 98, 101, 103, 107, 109, 112, 115, 118, 120, 123, 125, 128, 131, 134, 136, 139, 142, 145, 148, 151, 153
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 19 2020

Keywords

Crossrefs

Cf. A000720, A001221, A006590, A013939, A082767, A083399.

Programs

  • Mathematica
    Table[Sum[Ceiling[n/Prime[k]], {k, 1, n}], {n, 1, 60}]
Table[n + Sum[PrimeNu[k], {k, 1, n - 1}], {n, 1, 60}]
nmax = 60; CoefficientList[Series[x/(1 - x)^2 + (x/(1 - x)) Sum[x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
With[{nmax = 100}, Range[nmax] + Join[{0}, Accumulate[Table[PrimeNu[k], {k, 1, nmax - 1}]]]] (* Amiram Eldar, Sep 21 2024 *)
  • PARI
    a(n) = sum(k=1, n, ceil(n/prime(k))); \\ Michel Marcus, Feb 21 2020
  • PARI
    lista(nmax) = my(s = 1); for(n = 2, nmax, print1(s, ", "); s += omega(n-1) + 1); \\ Amiram Eldar, Sep 21 2024

Formula

G.f.: x/(1 - x)^2 + (x/(1 - x)) * Sum_{k>=1} x^prime(k) / (1 - x^prime(k)).
a(n) = n + Sum_{k=1..n-1} omega(k), where omega = A001221.
a(n) = n - omega(n) + Sum_{k=1..n} pi(floor(n/k)), where pi = A000720.
a(n) = n + A013939(n-1) for n >= 2. - Amiram Eldar, Sep 21 2024

A333494 a(1) = 1; a(n) = Sum_{k=1..n-1} ceiling(n/k) * a(k).

Original entry on oeis.org

1, 2, 7, 22, 69, 208, 634, 1903, 5734, 17210, 51702, 155107, 465561, 1396684, 4190689, 12572144, 37718360, 113155081, 339471195, 1018413586, 3055258062, 9165774828, 27497376189, 82492128568, 247476542954, 742429628932, 2227289352360, 6681868062822, 20045605585809
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 24 2020

Keywords

Crossrefs

Cf. A006590, A014668, A126656, A332846.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = Sum[Ceiling[n/k] a[k], {k, 1, n - 1}]; Table[a[n], {n, 1, 29}]
terms = 29; A[] = 0; Do[A[x] = x (1 + (1/(1 - x)) (A[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) + Sum_{k>=1} A(x^k))).
a(n) ~ c * 3^n, where c = 0.292080665386646518390576592052254840432101999262173908555857806023213143845... - Vaclav Kotesovec, Mar 25 2020
Previous Showing 21-25 of 25 results.

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