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 1-2 of 2 results.

A075885 a(n) = 1 + n + n*[n/2] + n*[n/2]*[n/3] + n*[n/2]*[n/3]*[n/4] +... where [x]=floor(x).

Original entry on oeis.org

1, 2, 5, 10, 29, 46, 169, 239, 745, 1450, 4111, 5182, 27157, 33164, 84001, 186496, 610065, 713474, 3061009, 3526553, 13783421, 27380452, 63264389, 71240523, 444872761, 620729126, 1400231613, 2615011102, 9094701085, 10008828958
Offset: 0

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Author

Paul D. Hanna, Oct 16 2002

Keywords

Comments

Conjecture: limit a(n)^(1/n) = L where L = 2.200161058099... is the geometric mean of Luroth expansions, where log(L) = Sum_{n>=1} log(n)/(n*(n+1)) = 0.7885305659115... (cf. A085361).
Compare the definition of a(n) to the exponential series:
exp(n) = 1 + n + n*(n/2) + n*(n/2)*(n/3) + n*(n/2)*(n/3)*(n/4) +...

Examples

			a(5) = 1 + 5 + 5[5/2] + 5[5/2][5/3] + 5[5/2][5/3][5/4] + 5[5/2][5/3][5/4][5/5] = 1 + 5 + 5*2 + 5*2*1 + 5*2*1*1 + 5*2*1*1*1 = 46.

Links

Crossrefs

Cf. A207643, A207644, A207645, A208060, A085361.

Programs

  • PARI
    {a(n)=1+sum(m=1,n,prod(k=1,m,floor(n/k)))}
for(n=0,60,print1(a(n),", "))
  • PARI
    a(n)=my(k=1);1+sum(m=1,n,k*=n\m) \\ Charles R Greathouse IV, Feb 20 2012

Formula

a(n) = 1 + Sum_{m=1..n} Product_{k=1..m} floor(n/k).

A331213 a(n) = 1 + Sum_{i=1..n} (-1)^i * Product_{j=1..i} floor(n/j).

Original entry on oeis.org

1, 0, 1, -2, 5, -4, 13, -27, 89, -80, 191, -450, 2365, -1182, 3221, -13034, 40433, -22320, 96373, -193761, 772981, -728930, 1599357, -3428425, 21411337, -13595724, 31407273, -110011850, 377746853, -198079308, 1096983421, -2241234465, 7565512161, -6472208192
Offset: 0

Views

Author

Seiichi Manyama, Jan 12 2020

Keywords

Comments

Compare to the exponential series: exp(-n) = 1 - n + n*(n/2) - n*(n/2)*(n/3) + n*(n/2)*(n/3)*(n/4) - ...

Examples

			a(4) = 1 - 4 + 4*floor(4/2) - 4*floor(4/2)*floor(4/3) + 4*floor(4/2)*floor(4/3)*floor(4/4) = 1 - 4 + 4*2 - 4*2*1 + 4*2*1*1 = 5.

Links

Crossrefs

Cf. similar sequences: A075885 (b=1), A208060 (b=2).
Cf. A010786.

Programs

  • Magma
    [1] cat [1+&+[(-1)^i*(&*[Floor(n/j):j in [1..i]]):i in [1..n]]:n in [1..33]]; // Marius A. Burtea, Jan 13 2020
  • Mathematica
    a[n_] := 1 + Sum[(-1)^i * Product[Floor[n/j], {j, 1, i}],{i, 1, n}]; Array[a, 34, 0] (* Amiram Eldar, Jan 13 2020 *)
  • PARI
    {a(n) = 1+sum(i=1, n, (-1)^i*prod(j=1, i, floor(n/j)))}
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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