A096386 Number of numbers <= n which are divisible by 2 or 3.

0, 0, 1, 2, 3, 3, 4, 4, 5, 6, 7, 7, 8, 8, 9, 10, 11, 11, 12, 12, 13, 14, 15, 15, 16, 16, 17, 18, 19, 19, 20, 20, 21, 22, 23, 23, 24, 24, 25, 26, 27, 27, 28, 28, 29, 30, 31, 31, 32, 32, 33, 34, 35, 35, 36, 36, 37, 38, 39, 39, 40, 40, 41, 42, 43, 43, 44, 44, 45, 46, 47, 47, 48, 48

Offset

0,4

Comments

First differences are 6-periodic.

References

Srinivasa Ramanujan, Question 723, Collected Papers of Srinivasa Ramanujan, p. 332, Ed. G. H. Hardy et al., AMS, Chelsea, 2000.

Links

Table of n, a(n) for n=0..73.

Michael Penn, A great floor equation by Ramanujan, YouTube video, 2021.

Srinivasa Ramanujan, Question 723, J. Ind. Math. Soc., Vol. 7 (1915), p. 240.

Index entries for two-way infinite sequences.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Formula

a(n) = [n/2] + [(n+3)/6] = [n/3] + [(n+2)/6] + [(n+4)/6].

G.f.: x^2*(1+x+x^2+x^4)/((1-x)*(1-x^6)).

a(-n) = -a(n-1)-1.

a(n) = 4 + a(n-6).

Sum_{n>=2} (-1)^n/a(n) = Pi/8 + log(2)/4. - Amiram Eldar, Jan 01 2025

Mathematica

CoefficientList[Series[x^2 (1 + x + x^2 + x^4)/((1 - x) (1 - x^6)), {x, 0, 73}], x] (* Michael De Vlieger, Apr 13 2016 *)
LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {0, 0, 1, 2, 3, 3, 4}, 80] (* Harvey P. Dale, Jul 04 2016 *)

Prog

(PARI) a(n) = floor(n/2) + floor((n+3)/6)
(PARI) a(n)=n\2+(n+3)\6

Crossrefs

Sequence in context: A057365 A014245 A350969 * A257063 A273663 A135671

Adjacent sequences: A096383 A096384 A096385 * A096387 A096388 A096389

Keyword

nonn,easy

Author

Ralf Stephan, Aug 05 2004

Extensions

New name based on Mar 13 2007 comment of Vladeta Jovovic. - Charles R Greathouse IV, Jan 10 2022

Status

approved