cp's OEIS Frontend

A033042 Sums of distinct powers of 5.

%I A033042 #86 Oct 30 2024 11:42:34
%S A033042 0,1,5,6,25,26,30,31,125,126,130,131,150,151,155,156,625,626,630,631,
%T A033042 650,651,655,656,750,751,755,756,775,776,780,781,3125,3126,3130,3131,
%U A033042 3150,3151,3155,3156,3250,3251,3255,3256,3275,3276,3280,3281,3750,3751
%N A033042 Sums of distinct powers of 5.
%C A033042 Numbers without any base-5 digits larger than 1.
%C A033042 a(n) modulo 2 is the Prouhet-Thue-Morse sequence A010060. - _Philippe Deléham_, Oct 17 2011
%C A033042 Values of k where A008977(k) does not end with 0. - _Henry Bottomley_, Nov 09 2022
%H A033042 T. D. Noe, <a href="/A033042/b033042.txt">Table of n, a(n) for n = 0..1023</a>
%H A033042 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, 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.]
%H A033042 K. Dilcher and L. Ericksen, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p24">Hyperbinary expansions and Stern polynomials</a>, Elec. J. Combin, 22, 2015, #P2.24.
%H A033042 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 A033042 N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%F A033042 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.
%F A033042 Numbers j such that the coefficient of x^j is > 0 in Product_{k>=0} (1 + x^(5^k)). - _Benoit Cloitre_, Jul 29 2003
%F A033042 a(n) = A097251(n)/4.
%F A033042 a(2n) = 5*a(n), a(2n+1) = a(2n)+1.
%F A033042 a(n) = Sum_{k>=0} A030308(n,k)*5^k. - _Philippe Deléham_, Oct 17 2011
%F A033042 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
%F A033042 G.f.: (1/(1 - x))*Sum_{k>=0} 5^k*x^(2^k)/(1 + x^(2^k)). - _Ilya Gutkovskiy_, Jun 04 2017
%p A033042 a:= proc(n) local m, r, b; m, r, b:= n, 0, 1;
%p A033042       while m>0 do r:= r+b*irem(m, 2, 'm'); b:= b*5 od; r
%p A033042     end:
%p A033042 seq(a(n), n=0..100);  # _Alois P. Heinz_, Mar 16 2013
%t A033042 t = Table[FromDigits[RealDigits[n, 2], 5], {n, 1, 100}]
%t A033042 (* _Clark Kimberling_, Aug 02 2012 *)
%t A033042 FromDigits[#,5]&/@Tuples[{0,1},7] (* _Harvey P. Dale_, May 22 2018 *)
%o A033042 (PARI) a(n) = subst(Pol(binary(n)), 'x, 5);
%o A033042 vector(50, i, a(i-1))  \\ _Gheorghe Coserea_, Sep 15 2015
%o A033042 (PARI) a(n)=fromdigits(binary(n),5) \\ _Charles R Greathouse IV_, Jan 11 2017
%o A033042 (Julia)
%o A033042 function a(n)
%o A033042     m, r, b = n, 0, 1
%o A033042     while m > 0
%o A033042         m, q = divrem(m, 2)
%o A033042         r += b * q
%o A033042         b *= 5
%o A033042     end
%o A033042 r end; [a(n) for n in 0:49] |> println # _Peter Luschny_, Jan 03 2021
%o A033042 (Python)
%o A033042 def A033042(n): return int(bin(n)[2:],5) # _Chai Wah Wu_, Oct 30 2024
%Y A033042 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.
%Y A033042 Cf. A000695, A005836, A008977, A010060, A033043-A033052.
%Y A033042 Row 5 of array A104257.
%K A033042 nonn,base,easy
%O A033042 0,3
%A A033042 _Clark Kimberling_
%E A033042 Extended by _Ray Chandler_, Aug 03 2004