A033043 Sums of distinct powers of 6.

0, 1, 6, 7, 36, 37, 42, 43, 216, 217, 222, 223, 252, 253, 258, 259, 1296, 1297, 1302, 1303, 1332, 1333, 1338, 1339, 1512, 1513, 1518, 1519, 1548, 1549, 1554, 1555, 7776, 7777, 7782, 7783, 7812, 7813, 7818, 7819, 7992, 7993, 7998, 7999, 8028, 8029, 8034

Offset

0,3

Comments

Numbers without any base-6 digits greater than 1.

Links

T. D. Noe, Table of n, a(n) for n = 0..1023

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. 6.

Formula

a(n) = Sum_{i=0..m} d(i)*6^i, where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.

a(n) = A097252(n)/5.

a(2n) = 6*a(n), a(2n+1) = a(2n)+1.

a(n) = Sum_{k>=0} A030308(n,k)*6^k. - Philippe Deléham, Oct 20 2011

G.f.: (1/(1 - x))*Sum_{k>=0} 6^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017

Mathematica

t = Table[FromDigits[RealDigits[n, 2], 6], {n, 0, 100}] (* Clark Kimberling, Aug 02 2012 *)
FromDigits[#, 6]&/@Tuples[{0, 1}, 6] (* Harvey P. Dale, Mar 31 2016 *)

Prog

(PARI) A033043(n, b=6)=subst(Pol(binary(n)), 'x, b) \\ M. F. Hasler, Feb 01 2016
(PARI) a(n)=fromdigits(binary(n), 6) \\ Charles R Greathouse IV, Jan 11 2017
(Julia)
function a(n)
    m, r, b = n, 0, 1
    while m > 0
        m, q = divrem(m, 2)
        r += b * q
        b *= 6
    end
r end; [a(n) for n in 0:46] |> println # Peter Luschny, Jan 03 2021

Crossrefs

Cf. A000695, A005836, A033043-A033052.

Row 6 of array A104257.

Sequence in context: A047190 A359530 A237711 * A037411 A025626 A175167

Adjacent sequences: A033040 A033041 A033042 * A033044 A033045 A033046

Keyword

nonn,base,easy

Author

Clark Kimberling

Extensions

Extended by Ray Chandler, Aug 03 2004

Status

approved