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

Author

Clark Kimberling

Keywords

• nonn
• base
• easy

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.

Crossrefs

Cf. A000695, A005836, A030308, A033043-A033052, A097252.

Row 6 of array A104257.

Programs

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

Maple

S:= {0,1}:
for i from 1 to 6 do S:= S union (S +~ 6^i) od:
sort(convert(S,list)); # Robert Israel, Apr 04 2025

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 *)

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

Python

def A033043(n): return int(bin(n)[2:],6) # Chai Wah Wu, Apr 04 2025

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

Extensions

Extended by Ray Chandler, Aug 03 2004