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A033042
Sums of distinct powers of 5.
38
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
0,3
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
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
Karl Dilcher and Larry Ericksen, Hyperbinary expansions and Stern polynomials, Elec. J. Combin, Vol. 22 (2015), Article 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.
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
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 *)
PROG
(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
(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
(Python)
def A033042(n): return int(bin(n)[2:], 5) # Chai Wah Wu, Oct 30 2024
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.
Row 5 of array A104257.
Sequence in context: A342610 A160529 A039572 * A039594 A137080 A025622
KEYWORD
nonn,base,easy
EXTENSIONS
Extended by Ray Chandler, Aug 03 2004
STATUS
approved