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.

371599983, 371599984, 371599985, 371599986, 371599987, 371599988, 371599989, 371599990, 371599991, 371599992, 500000000, 10000000000, 10371599983, 10371599984, 10371599985, 10371599986, 10371599987, 10371599988

Offset: 1

Author

Ryan Propper, Dec 10 2004

Keywords

• nonn
• base
• fini
• full

Comments

Related to a problem posed by Google and discussed on the MathWorld link.

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

Examples

			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.
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.
		

Links

• Daniel Hirschberg (dan(AT)ics.uci.edu), May 05 2007, Table of n, a(n) for n = 1..35
• Tanya Khovanova and Gregory Marton, Archive Labeling Sequences, arXiv:2305.10357 [math.HO], 2023. See p. 4.
• Mathworld, Problem 17 of Google Labs Aptitude Test Partially Answered, MathWorld Headline News, October 13 2004.

Crossrefs

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.

Extensions

More terms from Daniel Hirschberg (dan(AT)ics.uci.edu), May 05 2007