A033045 Sums of distinct powers of 8.

0, 1, 8, 9, 64, 65, 72, 73, 512, 513, 520, 521, 576, 577, 584, 585, 4096, 4097, 4104, 4105, 4160, 4161, 4168, 4169, 4608, 4609, 4616, 4617, 4672, 4673, 4680, 4681, 32768, 32769, 32776, 32777, 32832, 32833, 32840, 32841, 33280, 33281, 33288

Offset

0,3

Comments

Numbers without any base-8 digits greater than 1.

Every nonnegative n is a unique sum of the form a(p)+2a(q)+4a(r). This gives a one-to-one map of the set N_0 of all nonnegative integers to (N_0)^3. Furthermore, if, for a fixed positive integer m, to consider all sums of distinct powers of 2^m, then one can obtain a one-to-one map of the set N_0 to (N_0)^m. - Vladimir Shevelev, Nov 15 2008

Links

David A. Corneth, Table of n, a(n) for n = 0..9999 (first 1024 terms from T. D. Noe)

Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 45.

Michael Penn, "almost" a generating function..., YouTube video, 2020.

Formula

a(n) = Sum_{i=0..m} d(i)*8^i, where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.

a(n) = A097254(n)/7.

a(2n) = 8*a(n), a(2n+1) = a(2n)+1.

a(n) = Sum_{k>=0} A030308(n,k)*8^k. - Philippe Deléham, Oct 19 2011

G.f.: (1/(1 - x))*Sum_{k>=0} 8^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017

Example

a(7)=72 because 72_10 = 110_8.

Prog

(PARI) A033045(n, b=8)=subst(Pol(binary(n)), 'x, b) \\ M. F. Hasler, Feb 01 2016
(PARI) a(n) = fromdigits(binary(n), 8) \\ David A. Corneth, Dec 17 2020

Crossrefs

Cf. A000695, A005836, A033043-A033052.

Row 8 of array A104257.

Sequence in context: A152189 A042873 A342614 * A025633 A249697 A038287

Adjacent sequences: A033042 A033043 A033044 * A033046 A033047 A033048

Keyword

nonn,base,easy

Author

Clark Kimberling

Extensions

More terms from Patrick De Geest, Dec 23 2000

Status

approved