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A014778 Numbers k equal to the number of 1's in the decimal digits of all numbers <= k. 31

%I #50 Oct 02 2023 22:30:51

%S 0,1,199981,199982,199983,199984,199985,199986,199987,199988,199989,

%T 199990,200000,200001,1599981,1599982,1599983,1599984,1599985,1599986,

%U 1599987,1599988,1599989,1599990,2600000,2600001,13199998,35000000

%N Numbers k equal to the number of 1's in the decimal digits of all numbers <= k.

%C The full list of 84 terms is given in the b-file.

%C It can be proved that this sequence is finite. (The main idea of the proof is that the number of 1's used in positive integers <= k is greater than or equal to A(k) = (1/10)*(number of digits in positive integers from 1 to k) = (1/10) Sum_{i=1..k} (1+floor(log_10 i)). By considering the area below a logarithmic function and the corresponding integral, it can be shown that A(k)/k goes to infinity.) - _Joseph L. Pe_, Nov 05 2002

%C Fixed points of A094798. Sequence consists of six runs of ten consecutive numbers, ten pairs of consecutive numbers and four isolated numbers. - _David Wasserman_, Jun 29 2007

%D Maurice Protat, "Des Olympiades à l'Agrégation", Editions Ellipses, Paris 1997, p. 183.

%H Graeme McRae, May 26 2007, <a href="/A014778/b014778.txt">Table of n, a(n) for n = 1..84</a> (complete sequence)

%H Tanya Khovanova and Gregory Marton, <a href="https://arxiv.org/abs/2305.10357">Archive Labeling Sequences</a>, arXiv:2305.10357 [math.HO], 2023.

%H Ed Pegg Jr. and Eric W. Weisstein, <a href="http://mathworld.wolfram.com/news/2004-10-13/google/">Mathematica's Google Aptitude</a>, MathWorld Headline news, Oct 13 2004.

%e a(5)=199983 because the number of 1's in the decimal digits of the numbers from 0 to 199983 is 199983 and this is the 5th such number.

%t Join[{0},With[{nn=35*10^6},Position[Thread[{Accumulate[ DigitCount[ Range[nn],10,1]], Range[nn]}],{x_,x_}]]]//Flatten (* _Harvey P. Dale_, Oct 14 2017 *)

%o (Python)

%o from itertools import count, islice

%o def agen(s=0): # generator of terms

%o yield from (k for k in count(0) if (s:=s+str(k).count('1'))==k)

%o print(list(islice(agen(),26))) # _Michael S. Branicky_, Oct 02 2023

%Y Cf. A101639, A101640, A101641, A130427, A130428, A130429, A130430, A130431; cf. A130432 for the number of numbers in these sequences.

%Y Cf. A094798.

%Y Cf. A165617 for the sequence generalized to an arbitrary base. - Martin J. Erickson (erickson(AT)truman.edu), Oct 08 2010

%K base,fini,nonn,full

%O 1,3

%A Yves Babe, Maurice Protat, _Olivier Gérard_

%E Corrected and extended by Deepan Majmudar (deepan.majmudar(AT)hp.com), Nov 19 2004

%E 41 further terms from _Ryan Propper_, Dec 07 2004, who observed that there are no more terms <= 10^9

%E The final (84th) term 1111111110 was sent by Lambrecht Kok (L.P.Kok(AT)rug.nl), Jan 13 2005. He says: "H. van Haeringen and I showed that this list of 84 terms is complete on Dec 15 2004".

%E Independently shown to be complete by _Ryan Propper_ and Vaughan Pratt, Jan 08 2005

%E Edited by _M. F. Hasler_, Feb 12 2013

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Last modified March 19 03:33 EDT 2024. Contains 370952 sequences. (Running on oeis4.)