

A014778


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


31



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


COMMENTS

The full list of 84 terms is given in the bfile.
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



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)


CROSSREFS

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


KEYWORD

base,fini,nonn,full


AUTHOR



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


STATUS

approved



