A065712 -id:A065712 - cp's OEIS frontend

A027870 Number of zero digits in 2^n.

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

Offset: 0

N. J. A. Sloane

I conjecture that any value x = 0, 1, 2, ... occurs only a finite number of times N(x) = 36, 41, 31, 34, 25, 32, 37, 23, 43, 47, 33, ... in this sequence, for the last time at well defined indices i(x) = 86, 229, 231, 359, 283, 357, 475, 476, 649, 733, 648, ... - M. F. Hasler, Jul 09 2025

			2^31 = 2147483648 so a(31) = 0 and 2^42 = 4398046511104 so a(42) = 2.

Alois P. Heinz, Table of n, a(n) for n = 0..20000 (first 1001 terms from _Harry J. Smith_, Oct 27 2009)

Cf. A000079 (powers of 2), A007377 (2^n has no zeros).
Similar for other digits: A065712 (1's), A065710 (2's), A065714 (3's), A065715 (4's), A065716 (5's), A065717 (6's), A065718 (7's), A065719 (8's), A065744 (9's).
Cf. A031146 (index of first appearance of n in this sequence), A094776 (index of last occurrence of digit n in powers of 2).
Cf. A305932 (table with n in row a(n)).

a027870 = a055641 . a000079  -- Reinhard Zumkeller, Apr 30 2013

Table[ Count[ IntegerDigits[2^n], 0], {n, 0, 100} ]
DigitCount[2^Range[0,110],10,0] (* Harvey P. Dale, Nov 20 2011 *)

A027870(n)=#select(d->!d,digits(2^n)) \\ M. F. Hasler, Jun 14 2018

def A027870(n):
    return str(2**n).count('0') # Chai Wah Wu, Feb 14 2020

a(n) = A055641(A000079(n)). - Reinhard Zumkeller, Apr 30 2013

a(A007377(n)) = 0; A224782(n) <= a(n). - Reinhard Zumkeller, Apr 30 2013

Edited by M. F. Hasler, Jul 09 2025

A094776 a(n) = largest k such that the decimal representation of 2^k does not contain the digit n.

Original entry on oeis.org

86, 91, 168, 153, 107, 71, 93, 71, 78, 108

Offset: 0

Michael Taktikos, Jun 09 2004

These values are only conjectural.

The sequence could be extended to any nonnegative integer index n defining a(n) to be the largest k such that n does not appear as substring in the decimal expansion of 2^k. I conjecture that for n = 10, 11, 12, ... it continues (2000, 3020, 1942, 1465, 1859, 2507, 1950, 1849, 1850, ...). For example, curiously enough, the largest power of 2 in which the string "10" does not appear seems to be 2^2000. - M. F. Hasler, Feb 10 2023

			a(0) = 86 because 2^86 = 77371252455336267181195264 is conjectured to be the highest power of 2 that doesn't contain the digit 0.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 71, p. 25, Ellipses, Paris 2008.

Tanya Khovanova, 86 Conjecture, T. K.'s Math Blog, Feb. 15, 2011.
Popular Computing (Calabasas, CA), Two Tables, Vol. 1, (No. 9, Dec 1973), page PC9-16.

Cf. A259081 - A259083.
Cf. A027870 and A065712 - A065744 (number of '0's, ..., '9's in 2^n).
Cf. A034293 (numbers k such that 2^k has no '2').

f[n_] := Block[{a = {}, k = 1}, While[k < 10000, If[ Position[ Union[ IntegerDigits[ 2^k, 10]], n] == {}, AppendTo[a, k]]; k++ ]; a]; Table[ f[n][[ -1]], {n, 0, 9}] (* Robert G. Wilson v, Jun 12 2004 *)

A094776(n,L=10*20^#Str(n))={forstep(k=L, 0, -1, foreach(digits(1<M. F. Hasler, Feb 13 2023

def A094776(n, L=0):
   n = str(n)
   for k in range(L if L else 10*20**len(n), 0, -1):
      if n not in str(2**k): return k # M. F. Hasler, Feb 13 2023

A065710 Number of 2's in the decimal expansion of 2^n.

Original entry on oeis.org

0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 2, 2, 0, 2, 0, 0, 1, 1, 0, 2, 1, 1, 1, 1, 2, 1, 0, 0, 0, 1, 1, 0, 1, 4, 0, 3, 1, 2, 0, 1, 1, 3, 3, 3, 1, 2, 0, 1, 2, 1, 2, 2, 2, 3, 1, 3, 0, 2, 2, 3, 3, 2, 2, 4, 4, 4, 0, 1, 2, 4, 3, 1, 3, 6, 2, 0, 2, 4, 4, 4, 2, 3, 6, 2, 1, 5, 1, 2, 4, 4, 1, 2, 6

Offset: 0

Benoit Cloitre, Dec 04 2001

See A034293 for indices of zeros: It is conjectured that the last 0 appears at index 168 = A094776(2). More generally, I conjecture that any value x = 0, 1, 2, 3, ... occurs only a finite number of times N(x) = 23, 35, 28, 26, 41, 37, 34, 26, 34, 38, 33, 41, ... in this sequence, for the last time at a well defined index i(x) = 168, 176, 186, 268, 423, 361, 472, 555, 470, 562, 563, 735, .... - M. F. Hasler, Feb 10 2023, edited by M. F. Hasler, Jul 09 2025

			2^31 = 2147483648 so a(31) = 1.

M. F. Hasler, Table of n, a(n) for n = 0..10000 (first 1001 terms from Harry J. Smith), Feb 10 2023

Cf. 0's A027870, 1's A065712, 3's A065714, 4's A065715, 5's A065716, 6's A065717, 7's A065718, 8's A065719, 9's A065744.

Cf. A034293, A094776 (largest k for which 2^k has no digit n).

seq(numboccur(2, convert(2^n,base,10)),n=0..100); # Robert Israel, Jul 09 2025

Table[ Count[ IntegerDigits[2^n], 2], {n, 0, 100} ]

a(n) = #select(x->(x==2), digits(2^n)); \\ Michel Marcus, Jun 15 2018

def A065710(n):
    return str(2**n).count('2') # Chai Wah Wu, Feb 14 2020

a(n) = a(floor(n/10)) + [n == 2 (mod 10)], where [...] is the Iverson bracket. - M. F. Hasler, Feb 10 2023

More terms from Robert G. Wilson v, Dec 07 2001

A065715 Number of 4's in decimal expansion of 2^n.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 2, 1, 1, 0, 3, 0, 0, 2, 1, 1, 2, 0, 2, 3, 2, 1, 1, 1, 1, 2, 3, 1, 0, 0, 3, 0, 3, 1, 3, 2, 2, 2, 2, 1, 2, 2, 3, 0, 1, 2, 2, 3, 2, 1, 3, 1, 4, 3, 3, 2, 1, 0, 3, 4, 3, 4, 4, 0, 2, 2, 4, 3, 2, 3, 4, 3, 2, 0, 2, 4, 3, 3, 4, 5, 3, 2, 3, 1, 3, 1, 3, 3, 2

Offset: 0

Benoit Cloitre, Dec 04 2001

2^10 = 1024 so a(10)=1.

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. 0's A027870, 1's A065712, 2's A065710, 3's A065714, 5's A065716, 6's A065717, 7's A065718, 8's A065719, 9's A065744.

Table[ Count[ IntegerDigits[2^n], 4], {n, 0, 100} ]

a(n) = #select(x->(x==4), digits(2^n)); \\ Michel Marcus, Jun 15 2018

def A065715(n):
    return str(2**n).count('4') # Chai Wah Wu, Feb 14 2020

More terms from Robert G. Wilson v, Dec 07 2001

A065719 Number of 8's in decimal expansion of 2^n.

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Dec 04 2001

2^7 = 128 so a(7)=1.

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. 0's A027870, 1's A065712, 2's A065710, 3's A065714, 4's A065715, 5's A065716, 6's A065717, 7's A065718, 9's A065744.

Table[ Count[ IntegerDigits[2^n], 8], {n, 0, 100} ]
DigitCount[2^Range[0,100],10,8] (* Harvey P. Dale, Aug 02 2024 *)

a(n) = #select(x->(x==8), digits(2^n)); \\ Michel Marcus, Jun 15 2018

def A065719(n):
    return str(2**n).count('8') # Chai Wah Wu, Feb 14 2020

More terms from Robert G. Wilson v, Dec 07 2001

A065744 Number of 9's in the decimal expansion of 2^n.

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Dec 04 2001

See A035064 for the indices of zeros. I conjecture that any value x = 0, 1, 2, ... occurs only a finite number of times N(x) = 37, 27, 36, 46, 20, 31, 32, 30, 46, 29, 22, ... in this sequence, for the last time at well defined indices i(x) = 108, 197, 296, 277, 278, 315, 379, 555, 503, 504, 539, 696, 667, ... - M. F. Hasler, Jul 09 2025

			2^12 = 4096 so a(12)=1.

Harry J. Smith, Table of n, a(n) for n = 0..1000

Similar for other digits: A027870 (0's), A065712 (1's), A065710 (2's), A065714 (3's), A065715 (4's), A065716 (5's), A065717 (6's), A065718 (7's), A065719 (8's).

Cf. A035064 (2^n has no digit 9).

Table[ Count[ IntegerDigits[2^n], 9], {n, 0, 100} ]

Count(x,d)={ #select(t->t==d, digits(x)) }
a(n) = Count(2^n, 9) \\ Harry J. Smith, Oct 27 2009

def A065744(n):
    return str(2**n).count('9')  # Chai Wah Wu, Feb 14 2020

More terms from Robert G. Wilson v, Dec 07 2001

A065714 Number of 3's in decimal expansion of 2^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 3, 0, 1, 1, 1, 1, 1, 0, 1, 0, 4, 1, 3, 0, 1, 0, 1, 1, 1, 0, 3, 1, 3, 0, 2, 0, 1, 2, 0, 1, 2, 2, 0, 2, 3, 0, 4, 1, 3, 1, 4, 2, 1, 1, 1, 2, 3, 2, 3, 1, 2, 4, 1, 4, 3, 0, 3, 2, 3, 4, 4, 3, 3, 2, 1, 3, 0, 0, 4, 2, 2, 6, 1, 4, 4, 2

Offset: 0

Benoit Cloitre, Dec 04 2001

I conjecture that any value x = 0, 1, 2, ... occurs only a finite number of times N(x) = 34, 34, 24, 34, 39, 34, 35, 34, 35, 32, 33, 31, ... in this sequence, for the last time at well defined indices i(x) = 153, 139, 226, 237, 308, 386, 413, 506, 461, 578, 644, 732, 857, 657, 743, 768, 784, 848, 906, ... - M. F. Hasler, Jul 09 2025

			2^5 = 32 so a(5)=1.

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. A000079 (powers of 2), A035058 (2^n does not contain the digit 3).
Similar for other digits: A027870 (0's), A065712 (1's), A065710 (2's), this (3's), A065715 (4's), A065716 (5's), A065717 (6's), A065718 (7's), A065719 (8's), A065744 (9's).
Cf. A094776 (index of last occurrence of digit n in powers of 2).

Table[ Count[ IntegerDigits[2^n], 3], {n, 0, 100} ]

a(n) = #select(x->(x==3), digits(2^n)); \\ Michel Marcus, Jun 15 2018

def A065714(n):
    return str(2**n).count('3') # Chai Wah Wu, Feb 14 2020

More terms from Robert G. Wilson v, Dec 07 2001

A065716 Number of 5's in decimal expansion of 2^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 1, 1, 0, 0, 0, 2, 0, 0, 2, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 3, 1, 4, 1, 0, 1, 1, 0, 2, 1, 2, 1, 2, 2, 1, 1, 0, 2, 4, 1, 2, 2, 2, 0, 2, 2, 0, 0, 3, 5, 5, 1, 0, 1, 1, 3, 2, 4, 3, 3, 2, 1, 3, 3, 2, 1, 3, 4, 4, 3, 1, 1, 3, 4, 1, 3, 2, 4, 5, 5, 2, 2

Offset: 0

Benoit Cloitre, Dec 04 2001

			2^8 = 256 so a(8)=1.

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. 0's A027870, 1's A065712, 2's A065710, 3's A065714, 4's A065715, 6's A065717, 7's A065718, 8's A065719, 9's A065744.

Table[ Count[ IntegerDigits[2^n], 5], {n, 0, 100} ]
DigitCount[#,10,5]&/@(2^Range[0,100]) (* Harvey P. Dale, Nov 13 2021 *)

a(n) = #select(x->(x==5), digits(2^n)); \\ Michel Marcus, Jun 15 2018

def A065716(n):
    return str(2**n).count('5') # Chai Wah Wu, Feb 14 2020

More terms from Robert G. Wilson v, Dec 07 2001

A065717 Number of 6's in decimal expansion of 2^n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 0, 0, 1, 2, 0, 2, 0, 2, 1, 0, 1, 2, 0, 1, 1, 3, 0, 1, 0, 2, 0, 1, 1, 2, 0, 3, 0, 3, 1, 2, 1, 2, 0, 0, 3, 1, 0, 1, 2, 4, 1, 3, 1, 3, 1, 4, 2, 1, 1, 1, 3, 6, 1, 1, 3, 2, 2, 3, 2, 4, 1, 2, 4, 3, 6, 3, 2, 2, 4, 0, 1, 4, 0, 3, 4, 3, 3, 2, 3, 5

Offset: 0

Benoit Cloitre, Dec 04 2001

			2^8 = 256 so a(8)=1.

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. 0's A027870, 1's A065712, 2's A065710, 3's A065714, 4's A065715, 5's A065716, 7's A065718, 8's A065719, 9's A065744.

Table[ Count[ IntegerDigits[2^n], 6], {n, 0, 100} ]
DigitCount[#,10,6]&/@(2^Range[0,100]) (* Harvey P. Dale, Feb 15 2020 *)

a(n) = #select(x->(x==6), digits(2^n)); \\ Andrew Howroyd, Apr 25 2020

def A065717(n):
    return str(2**n).count('6') # Chai Wah Wu, Feb 14 2020

More terms from Robert G. Wilson v, Dec 07 2001

A065718 Number of 7's in decimal expansion of 2^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 3, 0, 1, 2, 0, 1, 2, 1, 1, 0, 2, 1, 3, 2, 3, 1, 3, 0, 0, 1, 1, 1, 4, 2, 3, 0, 0, 1, 2, 2, 0, 2, 4, 2, 3, 2, 1, 0, 2, 3, 3, 1, 3, 3, 2, 2, 2, 0, 1, 3, 2, 5, 3, 3, 2, 2, 3, 1, 3, 3, 1, 2, 4, 2, 2, 2, 2, 5, 2, 1, 2, 5, 2, 4, 4, 2, 3

Offset: 0

Benoit Cloitre, Dec 04 2001

			2^15 = 32768 so a(15)=1.

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. 0's A027870, 1's A065712, 2's A065710, 3's A065714, 4's A065715, 5's A065716, 6's A065717, 8's A065719, 9's A065744.

Table[ Count[ IntegerDigits[2^n], 7], {n, 0, 100} ]

a(n) = #select(x->(x==7), digits(2^n)); \\ Michel Marcus, Jun 15 2018

def A065718(n):
    return str(2**n).count('7') # Chai Wah Wu, Feb 14 2020

More terms from Robert G. Wilson v, Dec 07 2001

cp's OEIS Frontend

A027870 Number of zero digits in 2^n.

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A094776 a(n) = largest k such that the decimal representation of 2^k does not contain the digit n.

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A065710 Number of 2's in the decimal expansion of 2^n.

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A065715 Number of 4's in decimal expansion of 2^n.

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A065719 Number of 8's in decimal expansion of 2^n.

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A065744 Number of 9's in the decimal expansion of 2^n.

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