A027870 -id:A027870 - cp's OEIS frontend

A238939 Powers of 3 without the digit '0' in their decimal expansion.

1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 177147, 531441, 1594323, 4782969, 1162261467, 94143178827, 282429536481, 2541865828329, 7625597484987, 22876792454961, 617673396283947, 16677181699666569, 278128389443693511257285776231761

Offset: 1

M. F. Hasler, Mar 07 2014

Conjectured to be finite and complete. See the OEIS wiki page for further information, references and links.

M. F. Hasler, Zeroless powers, OEIS wiki, Mar 07 2014.

For the zeroless numbers (powers x^n), see A238938, A238939, A238940, A195948, A238936, A195908, A195946, A195945, A195942, A195943, A103662.
For the corresponding exponents, see A007377, A008839, A030700, A030701, A008839, A030702, A030703, A030704, A030705, A030706, A195944.
For other related sequences, see A052382, A027870, A102483, A103663.

Select[3^Range[0,100],DigitCount[#,10,0]==0&] (* Paolo Xausa, Oct 07 2023 *)

for(n=0,99,vecmin(digits(3^n))&& print1(3^n","))

a(n) = 3^A030700(n).

Keyword:fini removed by Jianing Song, Jan 28 2023 as finiteness is only conjectured.

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

A238936 Powers of 6 without the digit '0' in their decimal expansion.

Original entry on oeis.org

1, 6, 36, 216, 1296, 7776, 46656, 279936, 1679616, 2176782336, 16926659444736, 4738381338321616896, 36845653286788892983296, 17324272922341479351919144385642496

Offset: 1

M. F. Hasler, Mar 07 2014

Conjectured to be finite and complete. See the OEIS wiki page for further information, references and links.

M. F. Hasler, Zeroless powers, OEIS wiki, Mar 07 2014

Cf. A007377, A030700, A030701, A008839, A030702, A030703, A030705, A030706, A195944.
Cf. A195908, A195946, A195945.
Cf. A195942, A195943, A027870, A102483.

Select[6^Range[0,50],DigitCount[#,10,0]==0&] (* Harvey P. Dale, Dec 03 2020 *)

for(n=0,99,vecmin(digits(6^n))&& print1(6^n","))

a(n)=6^A030702(n).

Keyword:fini and keyword:full removed by Jianing Song, Jan 28 2023 as finiteness is only conjectured.

A238940 Powers of 4 without the digit '0' in their decimal expansion.

Original entry on oeis.org

1, 4, 16, 64, 256, 16384, 65536, 262144, 16777216, 268435456, 4294967296, 17179869184, 68719476736, 4722366482869645213696, 75557863725914323419136, 77371252455336267181195264

Offset: 1

M. F. Hasler, Mar 07 2014

Conjectured to be finite and complete. See the OEIS wiki page for further information, references and links.

M. F. Hasler, Zeroless powers, OEIS Wiki, Mar 07 2014

For the zeroless numbers (powers x^n), see A238938, A238939, A238940, A195948, A238936, A195908, A195946, A195945, A195942, A195943.
For the corresponding exponents, see A007377, A008839, A030700, A030701, A030702, A030703, A030704, A030705, A030706, A195944.
For other related sequences, see A052382, A027870, A102483.

Select[4^Range[0,50],DigitCount[#,10,0]==0&] (* Harvey P. Dale, Aug 31 2021 *)

for(n=0,99,vecmin(digits(4^n))&& print1(4^n","))

a(n)=4^A030701(n).

Keyword:fini removed by Jianing Song, Jan 28 2023 as finiteness is only conjectured.

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

cp's OEIS Frontend

A238939 Powers of 3 without the digit '0' in their decimal expansion.

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

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A238936 Powers of 6 without the digit '0' in their decimal expansion.

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A238940 Powers of 4 without the digit '0' in their decimal expansion.

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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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