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

Author

Clark Kimberling

Keywords

• nonn
• base
• easy

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

Examples

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

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.

Crossrefs

Cf. A000695, A005836, A033043-A033052.

Row 8 of array A104257.

Programs

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

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

Extensions

More terms from Patrick De Geest, Dec 23 2000