A130432 -id:A130432 - cp's OEIS frontend

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

Yves Babe, Maurice Protat, Olivier Gérard

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

			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.

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

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.

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

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 *)

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

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

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.

Original entry on oeis.org

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

Offset: 1

Ryan Propper, Dec 10 2004

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

			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.

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.

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.

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

A101641 Positive integers n for which n = f(n), where f(n) is the total number of 4's required when writing out all numbers between 0 and n.

Original entry on oeis.org

499999984, 499999985, 499999986, 499999987, 499999988, 499999989, 499999990, 499999991, 499999992, 499999993, 500000000, 10000000000, 10499999984, 10499999985, 10499999986, 10499999987, 10499999988, 10499999989

Offset: 1

Ryan Propper, Dec 11 2004

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 47 positive numbers n such that n is equal to the number of 4's in the decimal digits of all numbers <= n. - Daniel Hirschberg (dan(AT)ics.uci.edu), May 05 2007

			a(1) = 499999984, since writing out all numbers from 0 to 499999984 requires that 499999984 4's be used and since 499999984 is the first such positive integer.
a(4) = 499999987 because the number of 4's in the decimal digits of the numbers from 1 to 499999987 is 499999987 and this is the 4th such number.

Daniel Hirschberg (dan(AT)ics.uci.edu), May 05 2007, Table of n, a(n) for n = 1..47
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.

Cf. A014778 for proof these sequences are finite; Also A101639, A101640, A130427, A130428, A130429, A130430, A130431; cf. A130432 for the number of numbers in these sequences.

a(n) = 499999983 + n, n <= 10; a(n) = 500000000, n = 11

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

Keyword added by Charles R Greathouse IV, Jul 22 2010

A130428 List of numbers n such that n is equal to the number of 6's in the decimal digits of all numbers <= n.

Original entry on oeis.org

0, 9500000000, 9628399986, 9628399987, 9628399988, 9628399989, 9628399990, 9628399991, 9628399992, 9628399993, 9628399994, 9628399995, 10000000000, 19500000000, 19628399986, 19628399987, 19628399988, 19628399989

Offset: 1

Graeme McRae, May 26 2007

A finite sequence with 72 terms.

			a(5)=9628399988 because the number of 6's in the decimal digits of the numbers from 0 to 9628399988 is 9628399988 and this is the 5th such number.

Graeme McRae, May 26 2007, Table of n, a(n) for n = 1..72 (full sequence)
Tanya Khovanova and Gregory Marton, Archive Labeling Sequences, arXiv:2305.10357 [math.HO], 2023. See p. 4.

Cf. A014778 for proof these sequences are finite; Also A101639, A101640, A101641, A130427, A130429, A130430, A130431; Cf. A130432 for the number of numbers in these sequences.

A130429 List of all numbers n such that n is equal to the number of 7's in the decimal digits of all numbers <= n.

Original entry on oeis.org

0, 9465000000, 9471736170, 9500000000, 9757536170, 9965000000, 9971736170, 10000000000, 19465000000, 19471736170, 19500000000, 19757536170, 19965000000, 19971736170, 20000000000, 29465000000, 29471736170

Offset: 1

Graeme McRae, May 26 2007

A finite sequence with 49 terms.

			a(5)=9757536170 because the number of 7's in the decimal digits of the numbers from 0 to 9757536170 is 9757536170 and this is the 5th such number.

Graeme McRae, May 26 2007, Table of n, a(n) for n = 1..49 (full sequence)
Tanya Khovanova and Gregory Marton, Archive Labeling Sequences, arXiv:2305.10357 [math.HO], 2023. See p. 4.

Cf. A014778 for proof these sequences are finite; Also A101639, A101640, A101641, A130427, A130428, A130430, A130431; Cf. A130432 for the number of numbers in these sequences.

A130430 List of numbers n such that n is equal to the number of 8's in the decimal digits of all numbers <= n.

Original entry on oeis.org

0, 9465000000, 9486799989, 9486799990, 9486799991, 9486799992, 9486799993, 9486799994, 9486799995, 9486799996, 9486799997, 9497400000, 9498399989, 9498399990, 9498399991, 9498399992, 9498399993, 9498399994, 9498399995

Offset: 1

Graeme McRae, May 26 2007

A finite sequence with 344 terms.

			a(5)=9486799991 because the number of 8's in the decimal digits of the numbers from 0 to 9486799991 is 9486799991 and this is the 5th such number.

Graeme McRae, May 26 2007, Table of n, a(n) for n = 1..344 (full sequence)
Tanya Khovanova and Gregory Marton, Archive Labeling Sequences, arXiv:2305.10357 [math.HO], 2023. See p. 4.

Cf. A014778 for proof these sequences are finite; Also A101639, A101640, A101641, A130427, A130428, A130429, A130431; Cf. A130432 for the number of numbers in these sequences.

A101639 Positive integers n for which n = f(n), where f(n) is the total number of 2's required when writing out all numbers between 0 and n.

Original entry on oeis.org

28263827, 35000000, 242463827, 500000000, 528263827, 535000000, 10000000000, 10028263827, 10035000000, 10242463827, 10500000000, 10528263827, 10535000000

Offset: 1

Ryan Propper, Dec 10 2004

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

This is the complete list of all 13 positive numbers n such that n is equal to the number of 2's in the decimal digits of all numbers <= n. - Daniel Hirschberg (dan(AT)ics.uci.edu), May 05 2007

			a(1) = 28263827 since writing out all numbers from 0 to 28263827 requires that 28263827 2's be used and since 28263827 is the first such positive integer.
a(4) = 500000000 because the number of 2's in the decimal digits of the numbers from 1 to 500000000 is 500000000 and this is the 4th such number.

Tanya Khovanova and Gregory Marton, Archive Labeling Sequences, arXiv:2305.10357 [math.HO], 2023.
Mathworld, Problem 17 of Google Labs Aptitude Test Partially Answered, MathWorld Headline News, October 13 2004.

Cf. A014778 for proof these sequences are finite; Also A101640, A101641, A130427, A130428, A130429, A130430, A130431; cf. A130432 for the number of numbers in these sequences.

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

A130427 Complete list of all 5 numbers n such that n is equal to the number of 5's in the decimal digits of all numbers <= n.

Original entry on oeis.org

0, 10000000000, 20000000000, 30000000000, 40000000000

Offset: 1

Graeme McRae, May 26 2007

			a(5) = 40000000000 because the number of 5's in the decimal digits of the numbers from 0 to 40000000000 is 40000000000 and this is the 5th such number.

Tanya Khovanova and Gregory Marton, Archive Labeling Sequences, arXiv:2305.10357 [math.HO], 2023. See p. 4.

Cf. A014778 for proof these sequences are finite; Also A101639, A101640, A101641, A130428, A130429, A130430, A130431; Cf. A130432 for the number of numbers in these sequences.

A130431 Complete list of all 9 numbers n such that n is equal to the number of 9's in the decimal digits of all numbers <= n.

Original entry on oeis.org

0, 10000000000, 20000000000, 30000000000, 40000000000, 50000000000, 60000000000, 70000000000, 80000000000

Offset: 1

Graeme McRae, May 26 2007

			a(5)=40000000000 because the number of 9's in the decimal digits of the numbers from 0 to 40000000000 is 40000000000 and this is the 5th such number.

Tanya Khovanova and Gregory Marton, Archive Labeling Sequences, arXiv:2305.10357 [math.HO], 2023. See p. 4.

Cf. A014778 for proof these sequences are finite; Also A101639, A101640, A101641, A130427, A130428, A130429, A130430; Cf. A130432 for the number of numbers in these sequences.

A216400 Numbers k such that the number of times digit 'm' used for writing the decimal representation between 1 to k is equal to k for at least one value of m in the range m = 1 to 9.

Original entry on oeis.org

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, 28263827, 35000000, 35000001, 35199981

Offset: 1

V. Raman, Sep 06 2012

V. Raman, Table of n, a(n) for n = 1..590

Cf. A014778, A101639, A101640, A101641, A130427, A130428, A130429, A130430, A130431, A130432.

Union of A014778, A101639, A101640, A101641, A130427, A130428, A130429, A130430, A130431.

cp's OEIS Frontend

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

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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.

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A101641 Positive integers n for which n = f(n), where f(n) is the total number of 4's required when writing out all numbers between 0 and n.

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A130428 List of numbers n such that n is equal to the number of 6's in the decimal digits of all numbers <= n.

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A130429 List of all numbers n such that n is equal to the number of 7's in the decimal digits of all numbers <= n.

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A130430 List of numbers n such that n is equal to the number of 8's in the decimal digits of all numbers <= n.

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A101639 Positive integers n for which n = f(n), where f(n) is the total number of 2's required when writing out all numbers between 0 and n.

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A130427 Complete list of all 5 numbers n such that n is equal to the number of 5's in the decimal digits of all numbers <= n.

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A130431 Complete list of all 9 numbers n such that n is equal to the number of 9's in the decimal digits of all numbers <= n.

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