A065710 -id:A065710 - 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

A065712 Number of 1's in decimal expansion of 2^n.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 1, 1, 0, 2, 0, 1, 2, 0, 1, 2, 1, 0, 0, 3, 0, 1, 1, 0, 1, 3, 1, 3, 0, 3, 1, 1, 1, 2, 2, 2, 2, 0, 1, 3, 1, 0, 4, 4, 0, 3, 1, 3, 0, 3, 3, 0, 2, 2, 3, 6, 3, 1, 0, 2, 3, 3, 5, 1, 1, 5, 3, 1, 2, 5, 1, 4, 2, 2, 5, 2, 0, 5, 3, 1, 6, 2, 2, 4, 5, 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) = 26, 34, 30, 40, 26, 33, 39, 30, 30, 30, 38, ... in this sequence, for the last time at well defined indices i(x) = 91, 152, 185, 412, 245, 505, 346, 422, 499, 565, 529, 575, ... - M. F. Hasler, Jul 09 2025

			2^17 = 131072 so a(17) = 2.

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

Cf. A027870 (0'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).

Indices of zeros are listed in A035057 (2^n does not contain the digit 1).

Table[ Count[ IntegerDigits[2^n], 1], {n, 0, 100} ]
Table[DigitCount[2^n,10,1],{n,0,120}] (* Harvey P. Dale, Aug 15 2014 *)

a(n) = #select(x->(x==1), digits(2^n)); \\ Michel Marcus, Jun 15 2018

def A065712(n):
    return str(2**n).count('1') # Chai Wah Wu, Feb 14 2020

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

A034293 Numbers k such that 2^k does not contain the digit 2 (probably finite).

Original entry on oeis.org

0, 2, 3, 4, 6, 12, 14, 16, 20, 22, 23, 26, 34, 35, 36, 39, 42, 46, 54, 64, 74, 83, 168

Offset: 1

Erich Friedman

Is 168 the last term?
First row of A136291. - R. J. Mathar Apr 29 2008
Equivalently, indices of zeros in A065710. - M. F. Hasler, Feb 10 2023

			Here is 2^168, conjecturally the largest power of 2 that does not contain a 2: 374144419156711147060143317175368453031918731001856. - _N. J. A. Sloane_, Feb 10 2023

Cf. A007377.
See also similar sequences listed in A035064.
Cf. A065710 (number of '2's in 2^n), A094776.

[n: n in [0..1000] | not 2 in Intseq(2^n) ]; // Vincenzo Librandi, May 07 2015

isA034293 := proc(n) RETURN(not 2 in convert(2^n,base,10)) ; end: for n from 0 to 100000 do if isA034293(n) then print(n) ; fi ; od: # R. J. Mathar, Oct 04 2007

Join[{0}, Select[ Range@10000, FreeQ[ IntegerDigits[2^# ], 2] &]] (* Shyam Sunder Gupta, Sep 01 2007 *)(* adapted by Vincenzo Librandi, May 07 2015 *)
Select[Range[0, 10^4], DigitCount[2^#][[2]] == 0 &] (* Michael De Vlieger, Apr 29 2016 *)

is(n)=setsearch(Set(digits(2^n)),2)==0 \\ Charles R Greathouse IV, May 10 2016

is_A034293(n)=!foreach(digits(2^n),d,d==2&&return) \\ M. F. Hasler, Feb 10 2023

def is_A034293(n): return'2'not in str(2**n)
[n for n in range(199) if is_A034293(n)] # M. F. Hasler, Feb 10 2023

The last term is A094776(2), by definition. - M. F. Hasler, Feb 10 2023

Edited by N. J. A. Sloane, Oct 03 2007

Removed keyword "fini" since it is only a conjecture that this sequence contains only finitely many terms. - Altug Alkan, May 07 2016

cp's OEIS Frontend

A027870 Number of zero digits in 2^n.

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A065712 Number of 1's in 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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A065714 Number of 3's in decimal expansion of 2^n.

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