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

0, 1, 199981, 199982, 199983, 199984, 199985, 199986, 199987, 199988, 199989, 199990, 200000, 200001, 1599981, 1599982, 1599983, 1599984, 1599985, 1599986, 1599987, 1599988, 1599989, 1599990, 2600000, 2600001, 13199998, 35000000

Offset

1,3

Comments

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

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

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

References

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

Links

Graeme McRae, May 26 2007, Table of n, a(n) for n = 1..84 (complete sequence)

Tanya Khovanova and Gregory Marton, Archive Labeling Sequences, arXiv:2305.10357 [math.HO], 2023.

Ed Pegg Jr. and Eric W. Weisstein, Mathematica's Google Aptitude, MathWorld Headline news, Oct 13 2004.

Example

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.

Mathematica

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 *)

Prog

(Python)
from itertools import count, islice
def agen(s=0): # generator of terms
    yield from (k for k in count(0) if (s:=s+str(k).count('1'))==k)
print(list(islice(agen(), 26))) # Michael S. Branicky, Oct 02 2023

Crossrefs

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

Cf. A094798.

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

Sequence in context: A126558 A339752 A216400 * A094799 A163500 A164321

Adjacent sequences: A014775 A014776 A014777 * A014779 A014780 A014781

Keyword

base,fini,nonn,full

Author

Yves Babe, Maurice Protat, Olivier Gérard

Extensions

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

41 further terms from Ryan Propper, Dec 07 2004, who observed that there are no more terms <= 10^9

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".

Independently shown to be complete by Ryan Propper and Vaughan Pratt, Jan 08 2005

Edited by M. F. Hasler, Feb 12 2013

Status

approved