A178501 -id:A178501 - cp's OEIS frontend

A011557 Powers of 10: a(n) = 10^n.

1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 100000000000, 1000000000000, 10000000000000, 100000000000000, 1000000000000000, 10000000000000000, 100000000000000000, 1000000000000000000

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 10), L(1, 10), P(1, 10), T(1, 10). Essentially same as Pisot sequences E(10, 100), L(10, 100), P(10, 100), T(10, 100). See A008776 for definitions of Pisot sequences.
Same as k^n in base k. - Dominick Cancilla, Aug 02 2010 [Corrected by Jianing Song, Sep 17 2022]
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 10-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
Smallest n+1 digit number greater than 0 (with offset 0). - Wesley Ivan Hurt, Jan 17 2014
Numbers with digit sum = 1, or, A007953(a(n)) = 1. - Reinhard Zumkeller, Jul 17 2014
Does not satisfy Benford's law. - N. J. A. Sloane, Feb 14 2017

Philip Morrison et al., Powers of Ten, Scientific American Press, 1982 and later editions.
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

T. D. Noe, Table of n, a(n) for n = 0..100
Kees Boeke, Cosmic View: The Universe in 40 Jumps (1957) [The original "powers of ten" book]
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Charles and Ray Eames, Powers of Ten
Tanya Khovanova, Recursive Sequences
Robert Price, Comments on A011557 concerning Elementary Cellular Automata, Feb 21 2016
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Science, Optics and You, Secret Worlds: The Universe Within [Powers of Ten]
Eric Weisstein's World of Mathematics, 10
Eric Weisstein's World of Mathematics, Digitaddition
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Wikipedia, Powers of Ten
S. Wolfram, A New Kind of Science
Index entries for linear recurrences with constant coefficients, signature (10).
Index to Elementary Cellular Automata
Index entries for sequences related to cellular automata
Index entries for sequences related to Benford's law

Cf. A000005, A000012, A000290, A000400, A007953, A008776, A159991, A178500, A242614.
Cf. A178501: this sequence with 0 prefixed.
Row 5 of A329332.

a011557 = (10 ^)
a011557_list = iterate (* 10) 1
-- Reinhard Zumkeller, Jul 05 2013, Feb 05 2012

A011557:=n->10^n; seq(A011557(n), n=0..40); # Wesley Ivan Hurt, Jan 17 2014

Table[10^n,{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)
10^Range[0,20] (* Harvey P. Dale, Sep 17 2023 *)

A011557(n):=10^n$ makelist(A011557(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=10^n \\ Charles R Greathouse IV, Jun 15 2011

print([10**n for n in range(19)]) # Michael S. Branicky, Jan 10 2021

a(n) = 10^n.
a(n) = 10*a(n-1).
G.f.: 1/(1-10*x).
E.g.f.: exp(10*x).
A000005(a(n)) = A000290(n+1). - Reinhard Zumkeller, Mar 04 2007
a(n) = 60^n/6^n = A159991(n)/A000400(n). - Reinhard Zumkeller, May 02 2009
a(n) = A178501(n+1); for n > 0: a(n) = A178500(n). - Reinhard Zumkeller, May 28 2010
Sum_{n>0} 1/a(n) = 1/9 = A000012. - Stefano Spezia, Apr 28 2024

Links to "Powers of Ten" books and videos added by N. J. A. Sloane, Nov 07 2009

A029793 Numbers k such that k and k^2 have the same set of digits.

Original entry on oeis.org

0, 1, 10, 100, 1000, 4762, 4832, 10000, 10376, 10493, 11205, 12385, 12650, 14829, 22450, 23506, 24605, 26394, 34196, 36215, 47620, 48302, 48320, 49827, 64510, 68474, 71205, 72510, 72576, 74510, 74528, 79286, 79603, 79836, 94583, 94867, 96123, 98376

Offset: 1

Patrick De Geest

This sequence has density 1: almost all numbers k have all 10 digits in both k and k^2. - Franklin T. Adams-Watters, Jun 28 2011

			{0, 1, 3, 4, 9} = digits of a(10) = 10493 and of 10493^2 = 110103049;
{0, 1, 2, 5, 6} = digits of a(100) = 162025 and of 162025^2 = 26252100625;
{0, 1, 3, 4, 6, 7, 8} = digits of a(1000) = 1764380 and of 1764380^2 = 3113036784400;
{1, 2, 3, 4, 7, 8, 9} = digits of a(10000) = 14872239 and of 14872239^2 = 221183492873121.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A029795, A029797, A030091; A178501 is a subsequence, A054038, A171102, A232659-A232662.

import Data.List (nub, sort)
a029793 n = a029793_list !! (n-1)
a029793_list = filter (\x -> digs x == digs (x^2)) [0..]
   where digs = sort . nub . show
-- Reinhard Zumkeller, Jun 27 2011

[ n: n in [0..10^5] | Set(Intseq(n)) eq Set(Intseq(n^2)) ];  // Bruno Berselli, Jun 28 2011

seq(`if`(convert(convert(n,base,10),set) = convert(convert(n^2,base,10),set), n, NULL), n=0..100000); # Nathaniel Johnston, Jun 28 2011

digitSet[n_] := Union[IntegerDigits[n]]; Select[Range[0, 99000], digitSet[#] == digitSet[#^2] &] (* Jayanta Basu, Jun 02 2013 *)

isA029793(n)=Set(Vec(Str(n)))==Set(Vec(Str(n^2))) \\ Charles R Greathouse IV, Jun 28 2011

(0L to 99999L).filter(n => n.toString.toCharArray.toSet == (n * n).toString.toCharArray.toSet) // Alonso del Arte, Jan 19 2020

A178500 a(n) = 10^n * signum(n).

Original entry on oeis.org

0, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 100000000000, 1000000000000, 10000000000000, 100000000000000, 1000000000000000, 10000000000000000, 100000000000000000, 1000000000000000000, 10000000000000000000, 100000000000000000000

Offset: 0

Reinhard Zumkeller, May 28 2010

a(n-1) is the minimum difference between an n-digit number (written in base 10, nonzero leading digit) and the product of its digits. For n > 1, it is also a number meeting that bound. See A070565. - Devin Akman, Apr 17 2019

Michael De Vlieger, Table of n, a(n) for n = 0..999
Index entries for linear recurrences with constant coefficients, signature (10).

Cf. A000533, A011557, A057427, A061601, A070565, A109002, A155559, A178501.

Partial sums of A063945.

Array[10^#*Sign[#] &, 20, 0] (* Michael De Vlieger, Apr 21 2019 *)

a(n) = A011557(n)*A057427(n).
For n > 0, a(n) = A011557(n).
a(n) = 10*A178501(n).
a(n) = A000533(n) - 1.
A061601(a(n)) = A109002(n+1).
From Elmo R. Oliveira, Jul 21 2025: (Start)
G.f.: 10*x/(1-10*x).
E.g.f.: 2*exp(5*x)*sinh(5*x).
a(n) = 10*a(n-1) for n > 1. (End)

A258682 Remove from n^2 all digits of n from left to right, in decimal representation.

Original entry on oeis.org

0, 0, 4, 9, 16, 2, 3, 49, 64, 81, 0, 2, 44, 69, 96, 22, 25, 289, 324, 36, 40, 44, 484, 59, 576, 6, 76, 9, 74, 841, 90, 96, 104, 1089, 1156, 122, 129, 169, 1444, 1521, 160, 681, 176, 189, 1936, 202, 211, 2209, 230, 201, 20, 260, 704, 2809, 2916, 302, 313

Offset: 0

Reinhard Zumkeller, Jun 07 2015

a(A029783(n)) = A029783(n)^2; a(A189056(n)) < A189056(n)^2;

a(A178501(n)) = 0.

			.    n   n^2 |  a(n)
. -----------+------
.   20   400 |    40
.   21   441 |    44
.   22   484 |   484
.   23   529 |    59
.   24   576 |   576
.   25   625 |     6
.   26   676 |    76  not 67, as digits of n are to be removed
.   27   729 |     9                                  from left to right
.   28   784 |    74
.   29   841 |   841
.   30   900 |    90  .

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000290, A029783, A189056, A178501.

import Data.List (delete)
a258682 n = read ('0' : minus (show (n ^ 2)) (show n)) :: Int  where
   minus [] _  = []
   minus us [] = us
   minus (u:us) vs | elem u vs = minus us $ delete u vs
                   | otherwise = u : minus us vs

a[n_]:=Module[{idn=IntegerDigits[n],idnn=IntegerDigits[n^2],idn1},
idn1=DeleteCases[idn,m_/;MemberQ[Complement[idn,idnn],m]];
Do[If[First[Position[idnn,idn1[[i]]]]!= {},idnn=Delete[idnn,First[Position[idnn,idn1[[i]]]]]],
{i,1,Length[idn1]}];FromDigits[idnn]]//Quiet;
a/@Range[0,56] (* Ivan N. Ianakiev, Jun 11 2015 *)

A136859 Numbers k such that k and k^2 use only the digits 0, 1, 4 and 6.

Original entry on oeis.org

0, 1, 4, 10, 40, 100, 400, 1000, 4000, 10000, 40000, 100000, 400000, 1000000, 4000000, 10000000, 40000000, 100000000, 400000000, 1000000000, 4000000000, 10000000000, 40000000000, 100000000000, 400000000000, 1000000000000, 4000000000000, 10000000000000, 40000000000000, 100000000000000, 400000000000000

Offset: 1

Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008

Generated with DrScheme.
Are the formulas conjectured or proved? For example, the analogous sequence for {0,1,2,4} contains the sporadic solution 1010000104010000101. Clearly, if a(n) is in the sequence then 10*a(n) is also in the sequence. Is there any term that is not 0, 1, or 4 times a power of 10? - M. F. Hasler, Jan 26 2016
Answer: the formulas were merely conjectures. It appears that it is an open question as to whether there is any other type of term. - N. J. A. Sloane, Jan 29 2016
David W. Wilson has observed that the real number n = 2/3 = 0.66666... with n^2 = 4/9 = 0.44444... (almost) satisfies the requirement of this sequence. - N. J. A. Sloane, Jan 30 2016

			400000000000000^2 = 160000000000000000000000000000.

David W. Wilson, Table of n, a(n) for n = 1..61
Jonathan Wellons, Tables of Shared Digits [archived].

Cf. A000290, A178501.

clearQ[n_]:=Module[{dc = DigitCount[n]}, dc[[2]] == dc[[3]] == dc[[5]] == dc[[7]] == dc[[8]] == dc[[9]] == 0]
Select[Range[0, 2*10^6], clearQ[#]&&clearQ[#^2] &] (* Vincenzo Librandi, Feb 02 2016 *)

Conjectures from Philippe Deléham, Mar 11 2014: (Start)
G.f.: x^2*(1+4*x)/(1-10*x^2);
a(1) = 0, a(2) = 1, a(3) = 4, a(n) = 10*a(n-2) for n>3. (End)
This yields: a(n) = 4^(n mod 2)*A178501(floor(n/2)), where A178501(n) = floor(10^(k-1)). - M. F. Hasler, Nov 09 2017

Replaced formulas by conjectures, deleted b-file and computer program based on these conjectures. - N. J. A. Sloane, Jan 29 2016
M. F. Hasler, Jan 29 2016, reports that he has confirmed that the terms shown are complete up to a(31) = 400000000000000. - N. J. A. Sloane, Jan 30 2016
Extended b-file with complete values up to a(61). - David W. Wilson, Feb 01 2016

A257085 Numbers n such that the decimal expansions of both n and n^2 only use the digits 0..6.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 10, 11, 12, 15, 16, 20, 21, 25, 32, 34, 35, 40, 45, 46, 50, 51, 55, 56, 60, 65, 66, 100, 101, 102, 105, 106, 110, 111, 112, 115, 116, 120, 121, 125, 142, 145, 146, 150, 152, 155, 156, 160, 162, 200, 201, 204, 205, 206, 210, 211, 215, 216, 225, 231, 235, 245, 246, 250, 251, 252, 254, 255, 256

Offset: 1

Danny Rorabaugh, Apr 15 2015

			116 is in the list because 116 and 116^2 = 13456 do not use the digits 7, 8 or 9.
182 is not in the list because it uses the digit 8 (even though 182^2 = 33124 would be fine).
253 is not in the list because 253^2 = 6409 uses the digit 9.

Danny Rorabaugh, Table of n, a(n) for n = 1..10000

Cf. A178501 (0..1), A136808(0..2), A136809(0..3), A136810 (0..4), A257086 (0..5).

Cf. A136859, A256633.

Select[Range@ 256, Total@ Take[DigitCount[#], {7, 9}] == 0 && Total@ Take[DigitCount[#^2], {7, 9}] == 0 &] (* Michael De Vlieger, Apr 17 2015 *)

isok(n) = ((vecmax(digits(n)) <= 6) && (vecmax(digits(n^2)) <= 6)) || (n==0); \\ Michel Marcus, Feb 02 2016

A257086 Numbers n such that the decimal expansions of both n and n^2 only use the digits 0..5.

Original entry on oeis.org

0, 1, 2, 5, 10, 11, 12, 15, 20, 21, 32, 35, 45, 50, 55, 100, 101, 102, 105, 110, 111, 112, 115, 120, 145, 150, 152, 155, 200, 201, 205, 210, 211, 235, 320, 321, 332, 335, 350, 351, 450, 451, 452, 500, 501, 502, 505, 550, 1000, 1001, 1002, 1005, 1010, 1011, 1012, 1015, 1020, 1021, 1050, 1055, 1100

Offset: 1

Danny Rorabaugh, Apr 15 2015

			115 is in the list because 115 and 115^2 = 13225 do not use the digits 6, 7, 8, or 9.
121 is not in the list because 121^2 = 14641 uses the digit 6.
149 is not in the list because it uses the digit 9 (even though 149^2 = 22201 would be okay).

Danny Rorabaugh, Table of n, a(n) for n = 1..5000

Cf. A178501 (0..1), A136808(0..2), A136809(0..3), A136810 (0..4), A257085 (0..6).

Cf. A136817, A256631.

Select[Range@ 1100, Total@ Take[DigitCount[#], {6, 9}] == 0 && Total@ Take[DigitCount[#^2], {6, 9}] == 0 &] (* Michael De Vlieger, Apr 17 2015 *)

A305706 a(n) = smallest m such that the sum of digits of n^m is greater than n, or 0 if no such m exists.

Original entry on oeis.org

0, 0, 2, 2, 2, 2, 2, 2, 2, 3, 0, 4, 3, 2, 2, 3, 3, 2, 4, 3, 14, 4, 5, 4, 4, 5, 4, 5, 6, 6, 14, 5, 6, 5, 6, 6, 6, 5, 5, 6, 14, 7, 6, 8, 6, 7, 6, 8, 7, 7, 16, 6, 6, 8, 7, 8, 7, 7, 9, 7, 15, 8, 8, 7, 8, 8, 9, 8, 8, 8, 18, 12, 9, 9, 8, 9, 9, 9, 9, 9, 18, 10, 11, 11, 10, 11, 11, 10, 9, 10, 17, 11, 11, 10, 11, 10, 12, 12, 10, 11, 0

Offset: 0

Max Alekseyev, Jun 08 2018

a(n) = smallest m such that A007953(n^m) > n, or 0 if no such m exists.

If n is a term of A178501, then a(n) = 0. - Felix Fröhlich, Jun 10 2018

Cf. A007953, A038206, A178501, A305707.

a(n) = if(sumdigits(n) < 2, return(0), my(m=2); while(1, if(sumdigits(n^m) > n, return(m)); m++)) \\ Felix Fröhlich, Jun 10 2018

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A011557 Powers of 10: a(n) = 10^n.

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A029793 Numbers k such that k and k^2 have the same set of digits.

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A178500 a(n) = 10^n * signum(n).

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A258682 Remove from n^2 all digits of n from left to right, in decimal representation.

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A136859 Numbers k such that k and k^2 use only the digits 0, 1, 4 and 6.

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A257085 Numbers n such that the decimal expansions of both n and n^2 only use the digits 0..6.

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A257086 Numbers n such that the decimal expansions of both n and n^2 only use the digits 0..5.

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