A033045 - OEIS

%I #52 Oct 31 2022 02:09:15

%S 0,1,8,9,64,65,72,73,512,513,520,521,576,577,584,585,4096,4097,4104,

%T 4105,4160,4161,4168,4169,4608,4609,4616,4617,4672,4673,4680,4681,

%U 32768,32769,32776,32777,32832,32833,32840,32841,33280,33281,33288

%N Sums of distinct powers of 8.

%C Numbers without any base-8 digits greater than 1.

%C 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

%H David A. Corneth, <a href="/A033045/b033045.txt">Table of n, a(n) for n = 0..9999</a> (first 1024 terms from T. D. Noe)

%H Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="https://arxiv.org/abs/2210.10968">Identities and periodic oscillations of divide-and-conquer recurrences splitting at half</a>, arXiv:2210.10968 [cs.DS], 2022, p. 45.

%H Michael Penn, <a href="https://www.youtube.com/watch?v=e-Q_PpfC9do">"almost" a generating function...</a>, YouTube video, 2020.

%F 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.

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

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

%F a(n) = Sum_{k>=0} A030308(n,k)*8^k. - _Philippe Deléham_, Oct 19 2011

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

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

%o (PARI) A033045(n,b=8)=subst(Pol(binary(n)),'x,b) \\ _M. F. Hasler_, Feb 01 2016

%o (PARI) a(n) = fromdigits(binary(n), 8) \\ _David A. Corneth_, Dec 17 2020

%Y Cf. A000695, A005836, A033043-A033052.

%Y Row 8 of array A104257.

%K nonn,base,easy

%O 0,3

%A _Clark Kimberling_

%E More terms from _Patrick De Geest_, Dec 23 2000