A020858 -id:A020858 - cp's OEIS frontend

A000351 Powers of 5: a(n) = 5^n.

1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, 48828125, 244140625, 1220703125, 6103515625, 30517578125, 152587890625, 762939453125, 3814697265625, 19073486328125, 95367431640625, 476837158203125, 2384185791015625, 11920928955078125

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 5), L(1, 5), P(1, 5), T(1, 5). Essentially same as Pisot sequences E(5, 25), L(5, 25), P(5, 25), T(5, 25). See A008776 for definitions of Pisot sequences.
a(n) has leading digit 1 if and only if n = A067497 - 1. - Lekraj Beedassy, Jul 09 2002
With interpolated zeros 0, 1, 0, 5, 0, 25, ... (g.f.: x/(1 - 5*x^2)) second inverse binomial transform of Fibonacci(3n)/Fibonacci(3) (A001076). Binomial transform is A085449. - Paul Barry, Mar 14 2004
Sums of rows of the triangles in A013620 and A038220. - Reinhard Zumkeller, May 14 2006
Sum of coefficients of expansion of (1 + x + x^2 + x^3 + x^4)^n. a(n) is number of compositions of natural numbers into n parts less than 5. a(2) = 25 there are 25 compositions of natural numbers into 2 parts less than 5. - Adi Dani, Jun 22 2011
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 5-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
Numbers n such that sigma(5n) = 5n + sigma(n). In fact we have this theorem: p is a prime if and only if all solutions of the equation sigma(p*x) = p*x + sigma(x) are powers of p. - Jahangeer Kholdi, Nov 23 2013
From Doug Bell, Jun 22 2015: (Start)
Empirical observation: Where n is an odd multiple of 3, let x = (a(n) + 1)/9 and let y be the decimal expansion of x/a(n); then y*(x+1)/x + 1 = y rotated to the left.
Example:
  a(3) = 125;
  x = (125 + 1)/9 = 14;
  y = 112, which is the decimal expansion of 14/125 = 0.112;
  112*(14 + 1)/14 + 1 = 121 = 112 rotated to the left.
(End)
a(n) is the number of n-digit integers that contain only odd digits (A014261). - Bernard Schott, Nov 12 2022
Number of pyramids in the Sierpinski fractal square-based pyramid at the n-th step, while A279511 gives the corresponding number of vertices (see IREM link with drawings). - Bernard Schott, Nov 29 2022

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..100
O. M. Cain, The Exceptional Selfcondensability of Powers of Five, arXiv:1910.13829 [math.HO], 2019.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 270
IREM Paris-Nord, La pyramide de Sierpinski (in French).
Tanya Khovanova, Recursive Sequences
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Box Fractal
Index entries for linear recurrences with constant coefficients, signature (5).

Cf. A009969 (even bisection), A013710 (odd bisection), A005054 (first differences), A003463 (partial sums).
Cf. A006495, A006496, A159991, A001021.
Cf. A001076, A008776, A013620, A038220, A067497, A085449, A014261.
Sierpinski fractal square-based pyramid: A020858 (Hausdorff dimension), A279511 (number of vertices), this sequence (number of pyramids).

a000351 = (5 ^)
a000351_list = iterate (* 5) 1  -- Reinhard Zumkeller, Oct 31 2012

[5^n : n in [0..30]]; // Wesley Ivan Hurt, Sep 27 2016

[ seq(5^n,n=0..30) ];
A000351:=-1/(-1+5*z); # Simon Plouffe in his 1992 dissertation

Table[5^n, {n, 0, 30}] (* Stefan Steinerberger, Apr 06 2006 *)
5^Range[0, 30] (* Harvey P. Dale, Aug 22 2011 *)

makelist(5^n,n,0,20); /* Martin Ettl, Dec 27 2012 */

a(n)=5^n \\ Charles R Greathouse IV, Jun 10 2011

def a(n): return 5**n
print([a(n) for n in range(24)]) # Michael S. Branicky, Nov 12 2022

(List.fill(50)(5: BigInt)).scanLeft(1: BigInt)( * ) // Alonso del Arte, May 31 2019

a(n) = 5^n.
a(0) = 1; a(n) = 5*a(n-1) for n > 0.
G.f.: 1/(1 - 5*x).
E.g.f.: exp(5*x).
a(n) = A006495(n)^2 + A006496(n)^2.
a(n) = A159991(n) / A001021(n). - Reinhard Zumkeller, May 02 2009
From Bernard Schott, Nov 12 2022: (Start)
Sum_{n>=0} 1/a(n) = 5/4.
Sum_{n>=0} (-1)^n/a(n) = 5/6. (End)
a(n) = Sum_{k=0..n} C(2*n+1,n-k)*A000045(2*k+1). - Vladimir Kruchinin, Jan 14 2025

A020857 Decimal expansion of log_2(3).

Original entry on oeis.org

1, 5, 8, 4, 9, 6, 2, 5, 0, 0, 7, 2, 1, 1, 5, 6, 1, 8, 1, 4, 5, 3, 7, 3, 8, 9, 4, 3, 9, 4, 7, 8, 1, 6, 5, 0, 8, 7, 5, 9, 8, 1, 4, 4, 0, 7, 6, 9, 2, 4, 8, 1, 0, 6, 0, 4, 5, 5, 7, 5, 2, 6, 5, 4, 5, 4, 1, 0, 9, 8, 2, 2, 7, 7, 9, 4, 3, 5, 8, 5, 6, 2, 5, 2, 2, 2, 8, 0, 4, 7, 4, 9, 1, 8, 0, 8, 8, 2, 4

Offset: 1

N. J. A. Sloane

The fractional part of the binary logarithm of 3 * 2^n (A007283) is the same as that of any number of the form log_2 (A007283(n)) (e.g., log_2(192) = 7.5849625...). Furthermore, a necessary but not sufficient condition for a number to be Fibbinary (A003714) is that the fractional part of its binary logarithm does not exceed that of this number. - Alonso del Arte, Jun 22 2012
Log_2(3)-1 = 0.58496... is the exponent in n^(log_2(3)-1), the asymptotic growth rate of the number of odd coefficients in (1+x)^n mod 2 (Cf. Steven Finch ref.). - Jean-François Alcover, Aug 13 2014
Equals the Hausdorff dimension of the Sierpiński triangle. - Stanislav Sykora, May 27 2015
The complexity of Karatsuba algorithm for the multiplication of two n-digit numbers is O(n^log_2(3)). - Jianing Song, Apr 28 2019

			log_2(3) = 1.5849625007211561814537389439...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 24, 257.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.16, p. 145.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
E. G. Dunne, Pianos and Continued Fractions
Shalom Eliahou, Le problème 3n+1 : y a-t-il des cycles non triviaux? (III), Images des Mathématiques, CNRS, 2011 (in French).
Steven Finch, Pascal Sebah and Zai-Qiao Bai, Odd Entries in Pascal's Trinomial Triangle, arXiv:0802.2654 [math.NT], 2008, p. 1.
Karatsuba, The Complexity of Computations, Proceedings of the Steklov Institute of Mathematics, 1995: 169-183.
Youngik Lee, Numerical Approach on Collatz Conjecture, Preprints.org, Brown Univ., 2024. See p. 13.
Simon Plouffe, log(3)/log(2) to 10000 digits
A. M. Reiter, Determining the dimension of fractals generated by Pascal's triangle, Fibonacci Quart, 31(2):112-120, 1993.
Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant
Eric Weisstein's World of Mathematics, Pascal's Triangle
Eric Weisstein's World of Mathematics, Sierpiński Sieve
Wikipedia, Karatsuba algorithm
Wikipedia, Sierpinski triangle
Index entries for transcendental numbers

Cf. decimal expansion of log_2(m): this sequence, A020858 (m=5), A020859 (m=6), A020860 (m=7), A020861 (m=9), A020862 (m=10), A020863 (m=11), A020864 (m=12), A152590 (m=13), A154462 (m=14), A154540 (m=15), A154847 (m=17), A154905 (m=18), A154995 (m=19), A155172 (m=20), A155536 (m=21), A155693 (m=22), A155793 (m=23), A155921 (m=24).

Cf. A102525.

evalf(log[2](3), 100); # Bernard Schott, Feb 02 2023

RealDigits[Log[2, 3], 10, 100][[1]] (* Alonso del Arte, Jun 22 2012 *)

log(3)/log(2) \\ Michel Marcus, Jan 11 2016

Equals 1 / A102525. - Bernard Schott, Feb 02 2023

Comment generalized by J. Lowell, Apr 26 2014

A020862 Decimal expansion of log_2(10).

Original entry on oeis.org

3, 3, 2, 1, 9, 2, 8, 0, 9, 4, 8, 8, 7, 3, 6, 2, 3, 4, 7, 8, 7, 0, 3, 1, 9, 4, 2, 9, 4, 8, 9, 3, 9, 0, 1, 7, 5, 8, 6, 4, 8, 3, 1, 3, 9, 3, 0, 2, 4, 5, 8, 0, 6, 1, 2, 0, 5, 4, 7, 5, 6, 3, 9, 5, 8, 1, 5, 9, 3, 4, 7, 7, 6, 6, 0, 8, 6, 2, 5, 2, 1, 5, 8, 5, 0, 1, 3, 9, 7, 4, 3, 3, 5, 9, 3, 7, 0, 1, 5

Offset: 1

N. J. A. Sloane

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 55.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers.

Cf. decimal expansion of log_2(m): A020857 (m=3), A020858 (m=5), A020859 (m=6), A020860 (m=7), A020861 (m=9), this sequence, A020863 (m=11), A020864 (m=12), A152590 (m=13), A154462 (m=14), A154540 (m=15), A154847 (m=17), A154905 (m=18), A154995 (m=19), A155172 (m=20), A155536 (m=21), A155693 (m=22), A155793 (m=23), A155921 (m=24).

RealDigits[Log[2, 10], 10, 100][[1]] (* Vincenzo Librandi, Aug 29 2013 *)

log(10)/log(2) \\ Charles R Greathouse IV, Aug 06 2020

Equals 1+A020858. - R. J. Mathar, Oct 25 2008

Definition improved by J. Lowell, May 03 2014

A155921 Decimal expansion of log_2(24) = 3+log_2(3).

Original entry on oeis.org

4, 5, 8, 4, 9, 6, 2, 5, 0, 0, 7, 2, 1, 1, 5, 6, 1, 8, 1, 4, 5, 3, 7, 3, 8, 9, 4, 3, 9, 4, 7, 8, 1, 6, 5, 0, 8, 7, 5, 9, 8, 1, 4, 4, 0, 7, 6, 9, 2, 4, 8, 1, 0, 6, 0, 4, 5, 5, 7, 5, 2, 6, 5, 4, 5, 4, 1, 0, 9, 8, 2, 2, 7, 7, 9, 4, 3, 5, 8, 5, 6, 2, 5, 2, 2, 2, 8, 0, 4, 7, 4, 9, 1, 8, 0, 8, 8, 2, 4

Offset: 1

N. J. A. Sloane, Oct 30 2009

This is the third term in the sequence of real numbers discussed in A229168-A229170. - N. J. A. Sloane, Sep 28 2013

			4.5849625007211561814537389439478165087598144076924810604557...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_2(m): A020857 (m=3), A020858 (m=5), A020859 (m=6), A020860 (m=7), A020861 (m=9), A020862 (m=10), A020863 (m=11), A020864 (m=12), A152590 (m=13), A154462 (m=14), A154540 (m=15), A154847 (m=17), A154905 (m=18), A154995 (m=19), A155172 (m=20), A155536 (m=21), A155693 (m=22), A155793 (m=23), this sequence.

Cf. A229177.

RealDigits[Log[2,24],10,120][[1]] (* Harvey P. Dale, Dec 07 2011 *)

Equals 1 + A020864 = 2 + A020859 = 3 + A020857. - Jianing Song, Nov 16 2024

A154847 Decimal expansion of log_2 (17).

Original entry on oeis.org

4, 0, 8, 7, 4, 6, 2, 8, 4, 1, 2, 5, 0, 3, 3, 9, 4, 0, 8, 2, 5, 4, 0, 6, 6, 0, 1, 0, 8, 1, 0, 4, 0, 4, 3, 5, 4, 0, 1, 1, 2, 6, 7, 2, 8, 2, 3, 4, 4, 8, 2, 0, 6, 8, 8, 1, 2, 6, 6, 0, 9, 0, 6, 4, 3, 8, 6, 6, 9, 6, 5, 0, 9, 0, 4, 7, 3, 8, 2, 0, 6, 8, 2, 9, 7, 3, 4, 3, 1, 5, 1, 8, 4, 3, 6, 8, 4, 2, 7

Offset: 1

N. J. A. Sloane, Oct 30 2009

			4.0874628412503394082540660108104043540112672823448206881266...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_2(m): A020857 (m=3), A020858 (m=5), A020859 (m=6), A020860 (m=7), A020861 (m=9), A020862 (m=10), A020863 (m=11), A020864 (m=12), A152590 (m=13), A154462 (m=14), A154540 (m=15), this sequence, A154905 (m=18), A154995 (m=19), A155172 (m=20), A155536 (m=21), A155693 (m=22), A155793 (m=23), A155921 (m=24).

RealDigits[Log[2, 17], 10, 100][[1]] (* Vincenzo Librandi, Aug 29 2013 *)

A155172 Decimal expansion of log_2 (20).

Original entry on oeis.org

4, 3, 2, 1, 9, 2, 8, 0, 9, 4, 8, 8, 7, 3, 6, 2, 3, 4, 7, 8, 7, 0, 3, 1, 9, 4, 2, 9, 4, 8, 9, 3, 9, 0, 1, 7, 5, 8, 6, 4, 8, 3, 1, 3, 9, 3, 0, 2, 4, 5, 8, 0, 6, 1, 2, 0, 5, 4, 7, 5, 6, 3, 9, 5, 8, 1, 5, 9, 3, 4, 7, 7, 6, 6, 0, 8, 6, 2, 5, 2, 1, 5, 8, 5, 0, 1, 3, 9, 7, 4, 3, 3, 5, 9, 3, 7, 0, 1, 5

Offset: 1

N. J. A. Sloane, Oct 30 2009

Equals 2 + A020858 = 1 + A020862 = A016643 / A002162. - Michel Marcus, Jul 28 2013

			4.3219280948873623478703194294893901758648313930245806120547...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_2(m): A020857 (m=3), A020858 (m=5), A020859 (m=6), A020860 (m=7), A020861 (m=9), A020862 (m=10), A020863 (m=11), A020864 (m=12), A152590 (m=13), A154462 (m=14), A154540 (m=15), A154847 (m=17), A154905 (m=18), A154995 (m=19), this sequence, A155536 (m=21), A155693 (m=22), A155793 (m=23), A155921 (m=24).

RealDigits[Log[2, 20], 10, 100][[1]] (* Vincenzo Librandi, Aug 29 2013 *)

A020861 Decimal expansion of log_2(9).

Original entry on oeis.org

3, 1, 6, 9, 9, 2, 5, 0, 0, 1, 4, 4, 2, 3, 1, 2, 3, 6, 2, 9, 0, 7, 4, 7, 7, 8, 8, 7, 8, 9, 5, 6, 3, 3, 0, 1, 7, 5, 1, 9, 6, 2, 8, 8, 1, 5, 3, 8, 4, 9, 6, 2, 1, 2, 0, 9, 1, 1, 5, 0, 5, 3, 0, 9, 0, 8, 2, 1, 9, 6, 4, 5, 5, 5, 8, 8, 7, 1, 7, 1, 2, 5, 0, 4, 4, 5, 6, 0, 9, 4, 9, 8, 3, 6, 1, 7, 6, 4, 8

Offset: 1

N. J. A. Sloane

			3.16992500144231236290747788789563301751962881538496...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers.

Cf. decimal expansion of log_2(m): A020857 (m=3), A020858 (m=5), A020859 (m=6), A020860 (m=7), this sequence, A020862 (m=10), A020863 (m=11), A020864 (m=12), A152590 (m=13), A154462 (m=14), A154540 (m=15), A154847 (m=17), A154905 (m=18), A154995 (m=19), A155172 (m=20), A155536 (m=21), A155693 (m=22), A155793 (m=23), A155921 (m=24).

RealDigits[Log[9]/Log[2],10,120][[1]] (* Harvey P. Dale, Jul 08 2013 *)
RealDigits[Log2[9],10,120][[1]] (* Harvey P. Dale, May 21 2023 *)

log(9)/log(2) \\ Charles R Greathouse IV, Aug 06 2020

Equals 2 * A020857. - Amiram Eldar, Jun 25 2023

Definition improved by J. Lowell, May 03 2014

A020863 Decimal expansion of log_2(11).

Original entry on oeis.org

3, 4, 5, 9, 4, 3, 1, 6, 1, 8, 6, 3, 7, 2, 9, 7, 2, 5, 6, 1, 9, 9, 3, 6, 3, 0, 4, 6, 7, 2, 5, 7, 9, 2, 9, 5, 8, 7, 0, 3, 2, 3, 1, 5, 2, 5, 6, 8, 1, 7, 6, 8, 0, 7, 1, 3, 1, 2, 8, 0, 1, 6, 4, 5, 7, 2, 6, 3, 3, 0, 6, 1, 9, 7, 2, 0, 0, 1, 8, 3, 5, 2, 7, 0, 9, 4, 9, 1, 2, 9, 9, 2, 8, 6, 9, 0, 0, 4, 8

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_2(m): A020857 (m=3), A020858 (m=5), A020859 (m=6), A020860 (m=7), A020861 (m=9), A020862 (m=10), this sequence, A020864 (m=12), A152590 (m=13), A154462 (m=14), A154540 (m=15), A154847 (m=17), A154905 (m=18), A154995 (m=19), A155172 (m=20), A155536 (m=21), A155693 (m=22), A155793 (m=23), A155921 (m=24).

RealDigits[Log[2,11],10,120][[1]] (* Harvey P. Dale, Nov 05 2011 *)

log(11)/log(2) \\ Charles R Greathouse IV, Aug 06 2020

Definition improved by J. Lowell, May 03 2014

A020864 Decimal expansion of log(12)/log(2).

Original entry on oeis.org

3, 5, 8, 4, 9, 6, 2, 5, 0, 0, 7, 2, 1, 1, 5, 6, 1, 8, 1, 4, 5, 3, 7, 3, 8, 9, 4, 3, 9, 4, 7, 8, 1, 6, 5, 0, 8, 7, 5, 9, 8, 1, 4, 4, 0, 7, 6, 9, 2, 4, 8, 1, 0, 6, 0, 4, 5, 5, 7, 5, 2, 6, 5, 4, 5, 4, 1, 0, 9, 8, 2, 2, 7, 7, 9, 4, 3, 5, 8, 5, 6, 2, 5, 2, 2, 2, 8, 0, 4, 7, 4, 9, 1, 8, 0, 8, 8, 2, 4

Offset: 1

N. J. A. Sloane

			3.58496250072115618145373894394781650875981440.....

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_2(m): A020857 (m=3), A020858 (m=5), A020859 (m=6), A020860 (m=7), A020861 (m=9), A020862 (m=10), A020863 (m=11), this sequence, A152590 (m=13), A154462 (m=14), A154540 (m=15), A154847 (m=17), A154905 (m=18), A154995 (m=19), A155172 (m=20), A155536 (m=21), A155693 (m=22), A155793 (m=23), A155921 (m=24).

RealDigits[Log[2, 12], 10, 100][[1]] (* Vincenzo Librandi, Aug 29 2013 *)

log(12)/log(2) \\ Charles R Greathouse IV, May 15 2019

Equals 1 + A020859. - R. J. Mathar, Oct 25 2008

Equals 2 + A020857 = -1 + A155921. - Jianing Song, Nov 16 2024

A152590 Decimal expansion of log_2(13).

Original entry on oeis.org

3, 7, 0, 0, 4, 3, 9, 7, 1, 8, 1, 4, 1, 0, 9, 2, 1, 6, 0, 3, 9, 6, 8, 1, 2, 6, 5, 4, 2, 5, 6, 6, 9, 4, 7, 3, 3, 6, 2, 8, 4, 3, 6, 4, 0, 1, 7, 9, 1, 0, 3, 7, 3, 6, 9, 5, 3, 8, 4, 6, 3, 5, 2, 5, 8, 4, 2, 8, 5, 5, 1, 8, 6, 6, 3, 3, 0, 2, 5, 3, 0, 0, 1, 4, 7, 3, 7, 6, 5, 3, 0, 2, 8, 1, 1, 5, 4, 8, 9

Offset: 1

N. J. A. Sloane, Oct 28 2009

			3.7004397181410921603968126542566947336284364017910373695384...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_2(m): A020857 (m=3), A020858 (m=5), A020859 (m=6), A020860 (m=7), A020861 (m=9), A020862 (m=10), A020863 (m=11), A020864 (m=12), this sequence, A154462 (m=14), A154540 (m=15), A154847 (m=17), A154905 (m=18), A154995 (m=19), A155172 (m=20), A155536 (m=21), A155693 (m=22), A155793 (m=23), A155921 (m=24).

RealDigits[Log[2, 13], 10, 100][[1]] (* Vincenzo Librandi, Aug 29 2013 *)

log(13)/log(2) \\ Charles R Greathouse IV, Aug 06 2020

cp's OEIS Frontend

A000351 Powers of 5: a(n) = 5^n.

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A020857 Decimal expansion of log_2(3).

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A020862 Decimal expansion of log_2(10).

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A155921 Decimal expansion of log_2(24) = 3+log_2(3).

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A154847 Decimal expansion of log_2 (17).

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A155172 Decimal expansion of log_2 (20).

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A020861 Decimal expansion of log_2(9).

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