A033042 Sums of distinct powers of 5.

0, 1, 5, 6, 25, 26, 30, 31, 125, 126, 130, 131, 150, 151, 155, 156, 625, 626, 630, 631, 650, 651, 655, 656, 750, 751, 755, 756, 775, 776, 780, 781, 3125, 3126, 3130, 3131, 3150, 3151, 3155, 3156, 3250, 3251, 3255, 3256, 3275, 3276, 3280, 3281, 3750, 3751

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• base
• easy

Comments

Numbers without any base-5 digits larger than 1.

a(n) modulo 2 is the Prouhet-Thue-Morse sequence A010060. - Philippe Deléham, Oct 17 2011

Values of k where A008977(k) does not end with 0. - Henry Bottomley, Nov 09 2022

Links

• T. D. Noe, Table of n, a(n) for n = 0..1023
• David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
• K. Dilcher and L. Ericksen, Hyperbinary expansions and Stern polynomials, Elec. J. Combin, 22, 2015, #P2.24.
• 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.
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Crossrefs

For generating functions Product_{k>=0} (1 + a*x^(b^k)) for the following values of (a,b) see: (1,2) A000012 and A000027, (1,3) A039966 and A005836, (1,4) A151666 and A000695, (1,5) A151667 and A033042, (2,2) A001316, (2,3) A151668, (2,4) A151669, (2,5) A151670, (3,2) A048883, (3,3) A117940, (3,4) A151665, (3,5) A151671, (4,2) A102376, (4,3) A151672, (4,4) A151673, (4,5) A151674.

Cf. A000695, A005836, A008977, A010060, A033043-A033052.

Row 5 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 *= 5
    end
r end; [a(n) for n in 0:49] |> println # Peter Luschny, Jan 03 2021

Maple

a:= proc(n) local m, r, b; m, r, b:= n, 0, 1;
      while m>0 do r:= r+b*irem(m, 2, 'm'); b:= b*5 od; r
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 16 2013

Mathematica

t = Table[FromDigits[RealDigits[n, 2], 5], {n, 1, 100}]
(* Clark Kimberling, Aug 02 2012 *)
FromDigits[#,5]&/@Tuples[{0,1},7] (* Harvey P. Dale, May 22 2018 *)

PARI

a(n) = subst(Pol(binary(n)), 'x, 5);
vector(50, i, a(i-1))  \\ Gheorghe Coserea, Sep 15 2015

PARI

a(n)=fromdigits(binary(n),5) \\ Charles R Greathouse IV, Jan 11 2017

Python

def A033042(n): return int(bin(n)[2:],5) # Chai Wah Wu, Oct 30 2024

Formula

a(n) = Sum_{i=0..m} d(i)*5^i, where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.

Numbers j such that the coefficient of x^j is > 0 in Product_{k>=0} (1 + x^(5^k)). - Benoit Cloitre, Jul 29 2003

a(n) = A097251(n)/4.

a(2n) = 5*a(n), a(2n+1) = a(2n)+1.

a(n) = Sum_{k>=0} A030308(n,k)*5^k. - Philippe Deléham, Oct 17 2011

liminf a(n)/n^(log(5)/log(2)) = 1/4 and limsup a(n)/n^(log(5)/log(2)) = 1. - Gheorghe Coserea, Sep 15 2015

G.f.: (1/(1 - x))*Sum_{k>=0} 5^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017

Extensions

Extended by Ray Chandler, Aug 03 2004