cp's OEIS Frontend

A101640 Positive integers n for which n = f(n), where f(n) is the total number of 3's required when writing out all numbers between 0 and n.

%I A101640 #20 Feb 16 2025 08:32:55
%S A101640 371599983,371599984,371599985,371599986,371599987,371599988,
%T A101640 371599989,371599990,371599991,371599992,500000000,10000000000,
%U A101640 10371599983,10371599984,10371599985,10371599986,10371599987,10371599988
%N A101640 Positive integers n for which n = f(n), where f(n) is the total number of 3's required when writing out all numbers between 0 and n.
%C A101640 Related to a problem posed by Google and discussed on the MathWorld link.
%C A101640 Together with the b-file, this gives the complete list of all 35 positive numbers n such that n is equal to the number of 3's in the decimal digits of all numbers <= n. - Daniel Hirschberg (dan(AT)ics.uci.edu), May 05 2007
%H A101640 Daniel Hirschberg (dan(AT)ics.uci.edu), May 05 2007, <a href="/A101640/b101640.txt">Table of n, a(n) for n = 1..35</a>
%H A101640 Tanya Khovanova and Gregory Marton, <a href="https://arxiv.org/abs/2305.10357">Archive Labeling Sequences</a>, arXiv:2305.10357 [math.HO], 2023. See p. 4.
%H A101640 Mathworld, <a href="https://mathworld.wolfram.com/news/2004-10-13/google/">Problem 17 of Google Labs Aptitude Test Partially Answered</a>, MathWorld Headline News, October 13 2004.
%e A101640 a(1) = 371599983, since writing out all numbers from 0 to 371599983 requires that 371599983 3's be used and since 371599983 is the first such positive integer.
%e A101640 a(4) = 371599986 because the number of 3's in the decimal digits of the numbers from 1 to 371599986 is 371599986 and this is the 4th such number.
%Y A101640 Cf. A014778 for proof these sequences are finite; Also A101639, A101641, A130427, A130428, A130429, A130430, A130431; cf. A130432 for the number of numbers in these sequences.
%K A101640 nonn,base,fini,full
%O A101640 1,1
%A A101640 _Ryan Propper_, Dec 10 2004
%E A101640 More terms from Daniel Hirschberg (dan(AT)ics.uci.edu), May 05 2007