A067497 -id:A067497 - 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

A067488 Powers of 2 with initial digit 1.

Original entry on oeis.org

1, 16, 128, 1024, 16384, 131072, 1048576, 16777216, 134217728, 1073741824, 17179869184, 137438953472, 1099511627776, 17592186044416, 140737488355328, 1125899906842624, 18014398509481984, 144115188075855872, 1152921504606846976, 18446744073709551616

Offset: 1

Amarnath Murthy, Feb 09 2002

Also smallest n-digit power of 2.

For each range 10^(n-1) to 10^n-1 there exists exactly 1 power of 2 with first digit 1 (floor(log_10(a(n))) = n-1). As such, the density of this sequence relative to all powers of 2 (A000079) is log(2)/log(10) (0.301..., A007524), which is prototypical of Benford's Law. - Charles L. Hohn, Jul 23 2024

Muniru A Asiru, Table of n, a(n) for n = 1..993
Index to divisibility sequences

Cf. A000079, A067497, A055642,
Other initial digits: A067480, A067481, A067482, A067483, A067484, A067485, A067486, A067487, A074116.
Cf. A074117, A074118.

Filtered(List([0..60],n->2^n),i->ListOfDigits(i)[1]=1); # Muniru A Asiru, Oct 22 2018

[2^n: n in [0..100] | Intseq(2^n)[#Intseq(2^n)] eq 1]; // Vincenzo Librandi, Dec 31 2024

Select[2^Range[0, 70], First[IntegerDigits[#]] == 1 &]  (* Harvey P. Dale, Mar 14 2011 *)

a(n)=2^ceil((n-1)*log(10)/log(2)) \\ Charles R Greathouse IV, Apr 08 2012

(List.fill(50)(2: BigInt)).scanLeft(1: BigInt)( * ).filter(.toString.startsWith("1")) // _Alonso del Arte, Jan 16 2020

a(n) = 2^ceiling((n-1)*log(10)/log(2)). - Benoit Cloitre, Aug 29 2002
From Charles L. Hohn, Jun 09 2024: (Start)
a(n) = 2^A067497(n-1).
A055642(a(n)) = n. (End)

A123384 Number of bits in binary expansion of 10^n.

Original entry on oeis.org

1, 4, 7, 10, 14, 17, 20, 24, 27, 30, 34, 37, 40, 44, 47, 50, 54, 57, 60, 64, 67, 70, 74, 77, 80, 84, 87, 90, 94, 97, 100, 103, 107, 110, 113, 117, 120, 123, 127, 130, 133, 137, 140, 143, 147, 150, 153, 157, 160, 163, 167, 170, 173, 177, 180, 183, 187, 190, 193, 196

Offset: 0

Andrew Caldwell (spongebobpj(AT)yahoo.com), Nov 09 2006

Number of powers of 2 less than or equal to 10^n. - Peter Munn, Nov 13 2019

			a(3)=10 because 10^3 = 1111101000_2.
10^1 = 10 = 1010_2 has 4 digits.

Cf. A000079, A007524, A011557, A066343, A067497.

Row 1 of A253635.

A007524 := log[10](2.0) ; for n from 0 to 40 do printf("%d,", 1+floor(n/A007524)) ; od: # R. J. Mathar, Nov 12 2006
a:=n->nops(convert(10^n,base,2)): seq(a(n),n=0..70); # Emeric Deutsch, Mar 26 2007

a[n_]:=1 + Floor[n/Log10[2]]; Array[a,60,0] (* Stefano Spezia, Aug 31 2024 *)

a(n) = 1 + floor(n/A007524) = 1 + floor(n/log_10(2)). - R. J. Mathar, Nov 12 2006
a(n) = 1 + A066343(n). - R. J. Mathar, Mar 02 2007
a(n) = A067497(n) for n >= 1. - Georg Fischer, Nov 02 2018

More terms from Emeric Deutsch, Mar 26 2007

A066343 Beatty sequence for log_2(10).

Original entry on oeis.org

3, 6, 9, 13, 16, 19, 23, 26, 29, 33, 36, 39, 43, 46, 49, 53, 56, 59, 63, 66, 69, 73, 76, 79, 83, 86, 89, 93, 96, 99, 102, 106, 109, 112, 116, 119, 122, 126, 129, 132, 136, 139, 142, 146, 149, 152, 156, 159, 162, 166, 169, 172, 176, 179, 182, 186, 189, 192, 195, 199

Offset: 1

Vladeta Jovovic, Dec 15 2001

Number of positive integers <= 10^n that are divisible by no prime exceeding 2.
Maximum number of prime divisors of positive integers <= 10^n counted with multiplicity. - Martin Renner, Apr 04 2014
You wish to represent the rational number n/d in decimal notation, where n is an integer, d is a nonzero integer, and precision(d) represents the number of decimal digits in d. The decimal notation representation of n/d will either terminate or repeat with a repetend. If the decimal notation representation terminates then this sequence defines the maximum number of decimal digits to the right of the decimal point (after truncating trailing zeros) for a given precision of d ... floor(precision(d) * log_2(10)). - Michael T Howard, Jul 17 2017
Beatty complement of A066344. - Jianing Song, Jan 27 2019

Harry J. Smith, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Beatty Sequence

Cf. A066344, A067497, A129344.

Cf. A020862 (log_2(10)).

seq(floor(log[2](10)*n),n=1..60); # Martin Renner, Apr 04 2014

Table[ Floor[ n*Log[2, 10]], {n, 60}] (* Robert G. Wilson v, May 27 2005 *)

{ l=log(10)/log(2); for (n=1, 1000, a=floor(n*l); write("b066343.txt", n, " ", a) ) } \\ Harry J. Smith, Feb 11 2010

def A066343(n): return (5**n).bit_length()+n-1 # Chai Wah Wu, Sep 08 2024

a(n) = floor(n*log_2(10)).

A172404 Numbers k such that 3 is the first digit of 2^k.

Original entry on oeis.org

5, 15, 25, 35, 45, 55, 65, 75, 78, 85, 88, 95, 98, 108, 118, 128, 138, 148, 158, 168, 178, 181, 188, 191, 201, 211, 221, 231, 241, 251, 261, 271, 274, 281, 284, 291, 294, 304, 314, 324, 334, 344, 354, 364, 367, 374, 377, 384, 387, 397, 407, 417, 427, 437, 447, 457, 467, 470, 477, 480, 487, 490, 500

Offset: 1

David Radcliffe, Nov 20 2010

The asymptotic density of this sequence is log_10(4/3) = 0.124938... - Amiram Eldar, Jan 27 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 4200 terms from Muniru A Asiru)

Cf. A000079.

Cf. A067497, A067469, A367294, A363060, A367295, A330243, A367296, A097415.

Filtered([1..500],n->ListOfDigits(2^n)[1]=3); # Muniru A Asiru, Oct 17 2018

x := 1.; L := []; for n from 0 to 500 do if 3 < x and x < 4 then L := [op(L), n] end if; x := 2*x; if x > 10 then x := (1/10)*x end if end do; L;

Select[Range[1000], IntegerDigits[2^#][[1]] == 3 &]

isok(n) = digits(2^n)[1] == 3; \\ Michel Marcus, Oct 18 2018

ans, x = [], 1.
for n in range(501):
    if 3 < x < 4: ans.append(n)
    x = x*2
    if x > 10: x = x / 10
print(ans)

from itertools import islice
def A172404_gen(): # generator of terms
    a, b, c, l = 3, 4, 1, 0
    while True:
        if a<=c:
            if cA172404_list = list(islice(A172404_gen(),30)) # Chai Wah Wu, Nov 13 2023

A330243 Numbers k such that the first digit of the decimal expansion of 2^k is 7.

Original entry on oeis.org

46, 56, 66, 76, 86, 96, 149, 159, 169, 179, 189, 242, 252, 262, 272, 282, 292, 345, 355, 365, 375, 385, 438, 448, 458, 468, 478, 488, 531, 541, 551, 561, 571, 581, 634, 644, 654, 664, 674, 727, 737, 747, 757, 767, 777, 830, 840, 850, 860, 870, 923, 933, 943, 953

Offset: 1

Eder Vanzei, Dec 06 2019

The asymptotic density of this sequence is log_10(8/7) = 0.057991... - Amiram Eldar, Jan 27 2021

			70368744177664 = 2^46.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000030, A000079, A008952, A067469, A067497, A172404.

Select[Range[1000], Floor[2^# / 10^(Floor[# * Log10[2]])] == 7 &] (* Amiram Eldar, Dec 07 2019 *)
Select[Range[1000],IntegerDigits[2^#][[1]]==7&] (* or *) Select[Range[ 1000],NumberDigit[2^#,IntegerLength[2^#]-1]==7&] (* Harvey P. Dale, Aug 10 2021 *)

A330243_list = [n for n in range(10**3) if str(2**n)[0] == '7'] # Chai Wah Wu, Dec 12 2019

A363060 Numbers k such that 5 is the first digit of 2^k.

Original entry on oeis.org

9, 19, 29, 39, 49, 59, 69, 102, 112, 122, 132, 142, 152, 162, 172, 195, 205, 215, 225, 235, 245, 255, 265, 298, 308, 318, 328, 338, 348, 358, 391, 401, 411, 421, 431, 441, 451, 461, 494, 504, 514, 524, 534, 544, 554, 587, 597, 607, 617, 627, 637, 647, 657, 680, 690

Offset: 1

Ctibor O. Zizka, May 16 2023

The asymptotic density of this sequence is log_10(6/5) = 0.0791812... . - Amiram Eldar, May 16 2023

In base B we may consider numbers k such that some integer Y >= 1 forms the first digit(s) of X^k. For such numbers k the following inequality holds: log_B(Y) - floor(log_B(Y)) <= k*log_B(X) - floor(k*log_B(X)) < log_B(Y+1) - floor(log_B(Y+1)). The irrationality of log_B(X) is the necessary condition; see the Links section. Examples in the OEIS: B = 10, X = 2; Y = 1 (A067497), Y = 2 (A067469), Y = 3 (A172404).

			k = 9: the first digit of 2^9 = 512 is 5, thus k = 9 is a term.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
P. Bohl, Über ein in der Theorie der säkularen Störungen vorkommendes Problem, J. Reine Angew. Math. 135 (1909), 189-283.
Luboš Pick, On One Application of Irrationality in Search of the Seventh Heaven, Pokroky matematiky, fyziky a astronomie, vol. 67 (2022), 37-44.
Hermann Weyl, Über die gibbs’sche Erscheinung und verwandte Konvergenzphänomene, Rend. Circ. Matem. Palermo 30 (1910), 377-407.
Hermann Weyl, Über die Gleichverteilung von Zahlen mod. Eins., Mathematische Annalen 77 (1916), 313-352.

Cf. A000079, A008952, A018856.

Cf. A067497, A067469, A172404, A367294, A367295, A330243, A367296, A097415.

R:= NULL: count:= 0: t:= 1:
for k from 1 while count < 100 do
  t:= 2*t;
  if floor(t/10^ilog10(t)) = 5 then R:= R,k; count:= count+1 fi
od:
R; # Robert Israel, May 19 2023

Select[Range[700], IntegerDigits[2^#][[1]] == 5 &] (* Amiram Eldar, May 16 2023 *)

isok(k) = digits(2^k)[1] == 5; \\ Michel Marcus, May 16 2023

from itertools import count, islice
def A363060_gen(startvalue=1): # generator of terms >= startvalue
    m = 1<<(k:=max(startvalue,1))
    for i in count(k):
        if str(m)[0]=='5':
            yield i
        m <<= 1
A363060_list = list(islice(A363060_gen(),20)) # Chai Wah Wu, May 21 2023

A367296 Numbers k such that 8 is the first digit of 2^k.

Original entry on oeis.org

3, 13, 23, 33, 43, 106, 116, 126, 136, 146, 199, 209, 219, 229, 239, 302, 312, 322, 332, 342, 395, 405, 415, 425, 435, 498, 508, 518, 528, 538, 591, 601, 611, 621, 631, 684, 694, 704, 714, 724, 787, 797, 807, 817, 827, 880, 890, 900, 910, 920, 983, 993, 1003

Offset: 1

Martin Renner, Nov 12 2023

The asymptotic density of this sequence is log_10(9/8) = 0.051152...

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000079, A067497, A067469, A172404, A367294, A363060, A367295, A330243, A097415.

x := 1:
L := []:
for n from 0 to 10^3 do
  if 8 <= x and x < 9 then
    L := [op(L), n]
  fi;
  x := 2*x;
  if x > 10 then
    x := (1/10)*x fi;
od:
L;
# alternative:
select(t -> floor(2^t/10^ilog10(2^t))=8, [$1..10^4]); # Robert Israel, Nov 12 2024

Select[Range[1010], IntegerDigits[2^#][[1]] == 8 &] (* Amiram Eldar, Nov 12 2023 *)

from itertools import islice
def A367296_gen(): # generator of terms
    a, b, c, l = 8, 9, 1, 0
    while True:
        if a<=c:
            if cA367296_list = list(islice(A367296_gen(),30)) # Chai Wah Wu, Nov 13 2023

A129344 a(n) is the number of powers of 2 that have n decimal digits.

Original entry on oeis.org

4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 3, 4, 3, 3

Offset: 1

Tanya Khovanova, May 28 2007

Ignoring the first term, first differences of A066343. - Andrew Woods, Jun 10 2013

			a(1) is 4 because there are 4 one-digit powers of 2: 1, 2, 4, 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000

First differences of A067497.

Cf. A000079, A020862, A066343.

Table[Transpose[ Select[Table[{n, 2^n}, {n, 0, 310}], IntegerDigits[ #[[2]]][[1]] == 1 &]][[1]][[k]] - Transpose[ Select[Table[{n, 2^n}, {n, 0, 310}], IntegerDigits[ #[[2]]][[1]] == 1 &]][[1]][[k - 1]], {k, 2, 94}]
Join[{4}, Differences @ Table[Floor[n*Log2[10]], {n, 100}]] (* Amiram Eldar, Apr 09 2021 *)

a(n) = my(k=0, i=0); while(#Str(2^k)!=n, k++); while(#Str(2^k)==n, i++; k++); i \\ Felix Fröhlich, Jan 19 2016

def A129344(n): return -(m:=5**(n-1)).bit_length()+(5*m).bit_length()+1 if n>1 else 4 # Chai Wah Wu, Sep 08 2024

For n>1, a(n) = floor(n*L)-floor((n-1)*L) where L = log(10)/log(2). - Andrew Woods, Jun 10 2013

Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log_2(10) (A020862). - Amiram Eldar, Apr 09 2021

A367294 Numbers k such that 4 is the first digit of 2^k.

Original entry on oeis.org

2, 12, 22, 32, 42, 52, 62, 72, 82, 92, 105, 115, 125, 135, 145, 155, 165, 175, 185, 198, 208, 218, 228, 238, 248, 258, 268, 278, 288, 301, 311, 321, 331, 341, 351, 361, 371, 381, 394, 404, 414, 424, 434, 444, 454, 464, 474, 484, 497, 507, 517, 527, 537, 547

Offset: 1

Martin Renner, Nov 12 2023

The asymptotic density of this sequence is log_10(5/4) = 0.096910...

Cf. A000079, A067497, A067469, A172404, A363060, A367295, A330243, A367296, A097415.

x := 1:
L := []:
for n from 0 to 10^3 do
  if 4 <= x and x < 5 then
    L := [op(L), n]
  fi;
  x := 2*x;
  if x > 10 then
    x := (1/10)*x fi;
od:
L;

Select[Range[550], IntegerDigits[2^#][[1]] == 4 &] (* Amiram Eldar, Nov 12 2023 *)

cp's OEIS Frontend

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

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A067488 Powers of 2 with initial digit 1.

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A123384 Number of bits in binary expansion of 10^n.

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A066343 Beatty sequence for log_2(10).

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A172404 Numbers k such that 3 is the first digit of 2^k.

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A330243 Numbers k such that the first digit of the decimal expansion of 2^k is 7.

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