A027870 -id:A027870 - cp's OEIS frontend

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

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

A305932 Irregular table: row n >= 0 lists all k >= 0 such that the decimal representation of 2^k has n digits '0' (conjectured).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 15, 16, 18, 19, 24, 25, 27, 28, 31, 32, 33, 34, 35, 36, 37, 39, 49, 51, 67, 72, 76, 77, 81, 86, 10, 11, 12, 17, 20, 21, 22, 23, 26, 29, 30, 38, 40, 41, 44, 45, 46, 47, 48, 50, 57, 58, 65, 66, 68, 71, 73, 74, 75, 84, 85, 95, 96, 122, 124, 129, 130, 149, 151, 184, 43, 53, 61, 69, 70

Offset: 0

M. F. Hasler, Jun 14 2018

A partition of the nonnegative integers (the rows being the subsets).
Although it remains an open problem to provide a proof that the rows are complete (as are all terms of A020665), we can assume it for the purpose of this sequence.
Read as a flattened sequence, a permutation of the nonnegative integers.

			The table reads:
n \ k's
0 : 0, 1, ..., 9, 13, 14, 15, 16, 18, 19, 24, 25, 27, (...), 81, 86 (cf. A007377)
1 : 10, 11, 12, 17, 20, 21, 22, 23, 26, 29, 30, 38, 40, 41, 44, (...), 151, 184
2 : 42, 52, 54, 55, 56, 59, 60, 62, 63, 64, 78, 80, 82, 92, 107, (...), 171, 231
3 : 43, 53, 61, 69, 70, 83, 87, 89, 90, 93, 109, 112, 114, 115, (...), 221, 359
4 : 79, 91, 94, 97, 106, 118, 126, 127, 137, 139, 157, 159, 170, (...), 241, 283
5 : 88, 98, 99, 103, 104, 113, 120, 143, 144, 146, 152, 158, 160, (...), 343, 357
...
Column 0 is A031146: least k such that 2^k has n digits '0' in base 10.
Row lengths = number of powers of 2 with exactly n '0's = (36, 41, 31, 34, 25, 32, 37, 23, 43, 47, 33, 35, 29, 27, 27, 39, 34, 34, 28, 29, ...): not in the OEIS.
Largest number in row n = (86, 229, 231, 359, 283, 357, 475, 476, 649, 733, 648, 696, 824, 634, 732, 890, 895, 848, 823, 929, 1092, ...): not in the OEIS.
Row number of n = Number of '0's in 2^n = A027870: (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, ...).
Inverse permutation (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 37, 38, 10, 11, 12, 13, 39, 14, 15, 40, 41, 42, 43, 16, 17, 44, 18, 19, 45, 46, 20, 21, ...) is not in the OEIS.

M. F. Hasler, Zeroless powers, OEIS Wiki, March 2014.

Cf. A007377, A031146.
Sequence A027870 yields the row number of a given integer.
Cf. A305933 (analog for 3^n), A305924 (for 4^n), ..., A305929 (for 9^n).

mx = 1000; g[n_] := g[n] = DigitCount[2^n, 10, 0]; f[n_] := Select[Range@mx, g@# == n &]; Table[f@n, {n, 0, 4}] // Flatten (* Robert G. Wilson v, Jun 20 2018 *)

apply( A305932_row(n,M=200*(n+1))=select(k->A027870(k)==n,[0..M]), [0..20]) \\ A027870(k)=#select(d->!d, digits(2^k))

Row n = { k >= 0 | A027870(k) = n }.

A306112 Largest k such that 2^k has exactly n digits 0 (in base 10), conjectured.

Original entry on oeis.org

86, 229, 231, 359, 283, 357, 475, 476, 649, 733, 648, 696, 824, 634, 732, 890, 895, 848, 823, 929, 1092, 1091, 1239, 1201, 1224, 1210, 1141, 1339, 1240, 1282, 1395, 1449, 1416, 1408, 1616, 1524, 1727, 1725, 1553, 1942, 1907, 1945, 1870, 1724, 1972, 1965, 2075, 1983, 2114, 2257, 2256

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) is the largest term in A007377: exponents of powers of 2 without digit 0.

There is no proof for any of the terms, just as for any term of A020665 and many similar / related sequences. However, the search has been pushed to many magnitudes beyond the largest known term, and the probability of any of the terms being wrong is extremely small, cf., e.g., the Khovanova link.

M. F. Hasler, Zeroless powers, OEIS Wiki, March 2014, updated 2018.
T. Khovanova, The 86-conjecture, Tanya Khovanova's Math Blog, Feb. 2011.
W. Schneider, No Zeros, 2000, updated 2003. (On web.archive.org--see A007496 for a cached copy.)

Cf. A031146: least k such that 2^k has n digits 0 in base 10.
Cf. A305942: number of k's such that 2^k has n digits 0.
Cf. A305932: row n lists exponents of 2^k with n digits 0.
Cf. A007377: { k | 2^k has no digit 0 } : row 0 of the above.
Cf. A238938: { 2^k having no digit 0 }.
Cf. A027870: number of 0's in 2^n (and A065712, A065710, A065714, A065715, A065716, A065717, A065718, A065719, A065744 for digits 1 .. 9).
Cf. A102483: 2^n contains no 0 in base 3.

A306112_vec(nMax,M=99*nMax+199,x=2,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(x^k)),nMax)]=k);a[^-1]}

A224782 Length of longest run of consecutive zeros in decimal representation of 2^n.

Original entry on oeis.org

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, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1

Offset: 0

Reinhard Zumkeller, Apr 30 2013

a(n) <= A027870(n);
a(A007377(n)) = 0;
a(A006889(n)) = n and a(m) <> n for m < A006889(n).

Julian Havil, Impossible?: Surprising Solutions to Counterintuitive Conundrums, Princeton University Press 2008, chapter 15, p. 176ff

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

import Data.List (group)
a224782 n = a224782_list !! n
a224782_list = map (foldl h 0 . group . show) a000079_list where
   h x zs@(z:_) = if z == '0' then max x $ length zs else x

A240962 Number of zeros in the decimal expansion of n^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 10, 1, 2, 2, 3, 2, 2, 1, 2, 1, 21, 1, 0, 5, 2, 3, 6, 3, 1, 1, 32, 6, 5, 7, 7, 3, 3, 6, 8, 6, 42, 5, 6, 10, 10, 5, 11, 4, 12, 11, 53, 5, 6, 12, 10, 8, 11, 15, 9, 5, 64, 12, 15, 14, 16, 13, 12, 13, 9, 16, 79, 12, 16, 15, 12, 14, 15

Offset: 1

Anthony Sand, Aug 05 2014

			a(1) = zerocount(1^1) = zerocount(1) = 0.
a(8) = zerocount(8^8) = zerocount(16777216) = 0.
a(9) = zerocount(9^9) = zerocount(387420489) = 1.
a(10) = zerocount(10^10) = zerocount(10000000000) = 10.

Anthony Sand, Table of n, a(n) for n = 1..1000

Cf. A000312, A055641, A027870, A066588, A240963.

seq(numboccur(0,convert(n^n,base,10)), n=1 .. 100); # Robert Israel, Aug 05 2014

Map[Count[IntegerDigits[#^#], 0] &, Range[2, 100]] (* Michael De Vlieger, Aug 06 2014 *)

a(n) = my(d = digits(n^n)); sum(i=1, #d, ! d[i]); \\ Michel Marcus, Aug 10 2014

for n in range(1,10**3):
  print(str(n**n).count('0'),end=', ') # Derek Orr, Aug 05 2014

a(n) = A055641(A000312(n)). - Michel Marcus, Aug 07 2014

A298607 Powers of 2 with the digit '0' in their decimal expansion.

Original entry on oeis.org

1024, 2048, 4096, 131072, 1048576, 2097152, 4194304, 8388608, 67108864, 536870912, 1073741824, 274877906944, 1099511627776, 2199023255552, 4398046511104, 8796093022208, 17592186044416, 35184372088832, 70368744177664, 140737488355328, 281474976710656, 1125899906842624

Offset: 1

Alonso del Arte, Jan 22 2018

The complement, A238938, is conjectured to be finite. Furthermore, Khovanova (see link) believes 2^86 = 77371252455336267181195264 is the largest power of 2 not in this sequence.

			2^12 = 4096 contains one 0 in its decimal representation, hence 4096 is in the sequence.
2^13 = 8192 contains no 0's and is thus not in the sequence.

Tanya Khovanova, "86 Conjecture", Tanya Khovanova's Math Blog, February 15, 2011.

Cf. A007377, A027870.

Select[2^Range[0, 63], DigitCount[#, 10, 0] > 0 &]

lista(nn) = {for (n=0, nn, if (vecsearch(Set(digits(p=2^n)), 0), print1(p, ", ")););} \\ Michel Marcus, Mar 05 2018

A305930 Number of digits '0' in 3^n (in base 10).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Jun 22 2018

			3^10 = 59049 is the smallest power of 3 having a digit 0, so a(10) = 1 is the first nonzero term.

Cf. A027870 (analog for 2^k), A030700 (indices of zeros).
Cf. A063555: index of first appearence of n in this sequence.
Cf. A305933: table with n in row a(n).

Haskell
```
a305930 = a055641 . a000244
```

Table[ Count[ IntegerDigits[3^n], 0], {n, 0, 100} ]
DigitCount[3^Range[0,110],10,0]

apply( A305930(n)=#select(d->!d,digits(3^n)), [0..99])

a(n) = A055641(A000244(n)).

a(A030700(n)) = 0; a(A305934(n)) = 1; a(A305931(n)) >= 1; a(A305933(n,k)) = n.

A252482 Exponents n such that the decimal expansion of the power 12^n contains no zeros.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 10, 14, 20, 26

Offset: 1

M. F. Hasler, Dec 17 2014

Conjectured to be finite.

See A245853 for the actual powers 12^a(n).

M. F. Hasler, Zeroless powers, OEIS Wiki, Mar 07 2014
Eric Weisstein's World of Mathematics, Zero

For zeroless powers x^n, see A238938 (x=2), A238939, A238940, A195948, A238936, A195908, A245852, A240945 (k=9), A195946 (x=11), A245853, A195945; A195942, A195943, A103662.
For the corresponding exponents, see A007377, A030700, A030701, A008839, A030702, A030703, A030704, A030705, A030706, this sequence A252482, A195944.
For other related sequences, see A052382, A027870, A102483, A103663.

Select[Range[0,30],DigitCount[12^#,10,0]==0&] (* Harvey P. Dale, Apr 06 2019 *)

for(n=0,9e9,vecmin(digits(12^n))&&print1(n","))

A320601 Exponents of powers of two having a digit zero in decimal.

Original entry on oeis.org

10, 11, 12, 17, 20, 21, 22, 23, 26, 29, 30, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 73, 74, 75, 78, 79, 80, 82, 83, 84, 85, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105

Offset: 1

M. F. Hasler, Oct 16 2018

Complement of A007377. It is a long-standing open problem to show that this sequence contains all numbers > 86.

			The first term is a(1) = 10 since 2^10 = 1024 is the smallest power of 2 having a digit 0.

Cf. A000079, A007377, A027870, A055641.

for(n=1,199,vecmin(digits(2^n))||print1(n","))

{ n | A027870(n) > 0}, where A027870 = A055641 o A000079.

A330024 a(n) = floor(n/z) where z is the number of zeros in the decimal expansion of 2^n, and a(n)=0 when z=0.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 11, 12, 0, 0, 0, 0, 17, 0, 0, 20, 21, 22, 23, 0, 0, 26, 0, 0, 29, 30, 0, 0, 0, 0, 0, 0, 0, 38, 0, 40, 41, 21, 14, 44, 45, 46, 47, 48, 0, 50, 0, 26, 17, 27, 27, 28, 57, 58, 29, 30, 20, 31, 31, 32, 65, 66, 0, 68, 23, 23, 71, 0

Offset: 0

Metin Sariyar, Nov 27 2019

Is a(229)=229 the largest term?
a(8949)=41; is 8949 the largest n such that a(n) >= 41?
Is 79391 the largest n such that a(n) <= 30?
Is 30 <= a(n) <= 36 true for all n >= 713789?
Conjecture: For every sequence which can be named as "digit k appears m times in the decimal expansion of 2^n", the sequences are finite for 0 <= k <= 9 and any given m >= 0. Every digit from 0 to 9 are inclined to appear an equal number of times in the decimal expansion of 2^n as n increases.

			a(11) = 11 because 2^11 = 2048, there is 1 zero in 2048 and the integer part of 11/1 is 11.

Metin Sariyar, Table of n, a(n) for n = 0..32000

Cf. A007377, A027870.

a:=[0]; for n in [1..72] do z:=Multiplicity(Intseq(2^n),0); if z ne 0 then  Append(~a,Floor(n/z)); else Append(~a,0); end if; end for; a; // Marius A. Burtea, Nov 27 2019

f:= proc(n) local z;
  z:= numboccur(0,convert(2^n,base,10));
  if z = 0 then 0 else floor(n/z) fi
end proc:
map(f, [$1..100]); # Robert Israel, Nov 28 2019

Do[z=DigitCount[2^n,10,0];an=IntegerPart[n/z];If[z==0,Print[0],Print[an]],{n,0,8000}]

a(n) = my(z=#select(d->!d, digits(2^n))); if (z, n\z, 0); \\ Michel Marcus, Jan 07 2020

def A330024(n):
  z=str(2**n).count('0')
  return n//z if z else 0 # Pontus von Brömssen, Jul 24 2021

Conjecture: a(n) = 33 (= floor(10/log_10(2))) for all sufficiently large n. - Pontus von Brömssen, Jul 23 2021

cp's OEIS Frontend

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

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A305932 Irregular table: row n >= 0 lists all k >= 0 such that the decimal representation of 2^k has n digits '0' (conjectured).

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A306112 Largest k such that 2^k has exactly n digits 0 (in base 10), conjectured.

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A224782 Length of longest run of consecutive zeros in decimal representation of 2^n.

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A240962 Number of zeros in the decimal expansion of n^n.

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A298607 Powers of 2 with the digit '0' in their decimal expansion.

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A305930 Number of digits '0' in 3^n (in base 10).

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A252482 Exponents n such that the decimal expansion of the power 12^n contains no zeros.

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