A007377 -id:A007377 - cp's OEIS frontend

A305942 Number of powers of 2 having exactly n digits '0' (in base 10), conjectured.

36, 41, 31, 34, 25, 32, 37, 23, 43, 47, 33, 35, 29, 27, 27, 39, 34, 34, 28, 29, 31, 30, 38, 25, 35, 35, 36, 40, 32, 40, 43, 39, 32, 30, 30, 32, 36, 39, 23, 26, 31, 37, 27, 28, 33, 39, 28, 44, 34, 27, 43, 33, 27, 32, 31, 27, 27, 32, 35, 34, 36, 28, 32, 39, 38, 40, 28, 43, 38, 32, 22

Offset: 0

M. F. Hasler, Jun 21 2018

a(0) = 36 is the number of terms in A007377 and in A238938, which includes the power 2^0 = 1.
These are the row lengths of A305932. It remains an open problem to provide a proof that these rows are complete (as for all terms of A020665), but the search has been pushed to many orders of magnitude beyond the largest known term, and the probability of finding an additional term is vanishing, cf. Khovanova link.
The average of the first 100000 terms is ~33.219 with a minimum of 12 and a maximum of 61. - Hans Havermann, Apr 26 2020

Hans Havermann, Table of n, a(n) for n = 0..10000
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.)

Row lengths of A305932 (row n = exponents of 2^k with n '0's).
Cf. A007377 = {k | 2^k has no digit 0}; A238938: powers of 2 with no digit 0.
Cf. A298607: powers of 2 with the digit '0' in their decimal expansion.
Cf. A020665: largest k such that n^k has no digit 0 in base 10.
Cf. A031146: least k such that 2^k has n digits 0 in base 10.
Cf. A071531: least r such that n^r has a digit 0, in base 10.
Cf. A306112: largest k such that 2^k has n digits 0, in base 10.

A305942(n,M=99*n+199)=sum(k=0,M,#select(d->!d,digits(2^k))==n)

A305942_vec(nMax,M=99*nMax+199,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(2^k)),nMax)]++);a[^-1]}

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]}

A131613 Numbers k such that the decimal expansion of 3^k contains no 9.

Original entry on oeis.org

0, 1, 3, 4, 5, 7, 8, 11, 12, 16, 19, 20, 21, 29, 32, 56

Offset: 1

Shyam Sunder Gupta, Sep 01 2007

I conjecture that 56 is the last term.

Numbers k such that the decimal expansion of 3^k contains no m: A030700 (m=0), A131627 (m=1), A131625 (m=2), A131629 (m=3), A131618 (m=4), A131617 (m=5), A131616 (m=6), A131615 (m=7), A131614 (m=8), this sequence (m=9).

Cf. A007377.

[n: n in [0..1000] | not 9 in Intseq(3^n) ]; // Vincenzo Librandi, May 06 2015

Join[{0}, Select[ Range@10000, FreeQ[ IntegerDigits[3^# ], 9] &]]

Adapted Mma and initial 0 added by Vincenzo Librandi, May 06 2015

A130696 Numbers k such that 2^k does not contain all ten decimal digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 83, 85, 86, 90, 91, 92, 93, 99, 102, 107, 108, 153, 168

Offset: 1

Greg Dresden, Jul 10 2007

It is believed that 168 is the last number in this list; 2^168 is a 51-digit number that contains all the digits except (oddly enough) 2.

There are no more terms less than 10^10. - David Radcliffe, Apr 11 2019

			20 is in this list because 2^20 = 1048576, which doesn't contain all ten digits.
68 is the first number not in this list; 2^68 = 295147905179352825856 and this contains all ten digits.

Cf. A007377, A062518, A090493.

Complement of A130694.

A2 := {}; Do[If[Length[Union[ IntegerDigits[2^ n]]] != 10, A2 = Join[A2, {n}]], {n, 1, 3000}]; Print[A2]
Select[Range[10^6]-1,MemberQ[DigitCount[2^#],0]&] (* Hans Rudolf Widmer, Jun 23 2021 *)

hasalldigits(n) = #vecsort(digits(n), , 8)==10
is(n) = !hasalldigits(2^n) \\ Felix Fröhlich, Apr 11 2019

print([n for n in range(1000) if len(set(str(2**n))) < 10]) # David Radcliffe, Apr 11 2019

a(1) = 0 prepended by David Radcliffe, Apr 11 2019

A050723 Numbers k such that the decimal expansion of 2^k contains no pair of consecutive equal digits (probably finite).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 20, 21, 22, 28, 29, 30, 31, 32, 34, 35, 36, 37, 48, 54, 55, 56, 66, 67, 68, 69, 80, 87, 104, 126

Offset: 1

Patrick De Geest, Sep 15 1999

No further terms up to 100000. - T. D. Noe, Sep 18 2012

			2^126 = 85070591730234615865843651857942052864.

Cf. A043096, A007377, A046260, A046268.

q:= n-> (s-> andmap(i-> s[i]<>s[i+1], [$1..length(s)-1]))(""||(2^n)):
select(q, [$0..200])[];  # Alois P. Heinz, Mar 07 2024

Select[Range[0,10000],!MemberQ[Differences[IntegerDigits[2^#]],0]&] (* Harvey P. Dale, Dec 24 2011 *)

A131629 Numbers k such that the decimal expansion of 3^k contains no 3.

Original entry on oeis.org

0, 2, 3, 4, 6, 7, 8, 10, 11, 14, 19, 27, 28, 34, 40, 50, 55, 84

Offset: 1

Shyam Sunder Gupta, Sep 01 2007

I conjecture that 84 is the last term.

Cf. similar sequences listed in A131613.

Cf. A007377, A136291.

[n: n in [0..1000] | not 3 in Intseq(3^n) ]; // Vincenzo Librandi, May 06 2015

Join[{0}, Select[ Range@10000, FreeQ[ IntegerDigits[3^# ], 3] &]]

Initial 0 added and Mathematica code adapted by Vincenzo Librandi, May 06 2015

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

A112388 a(n) is the smallest prime p such that p^n contains every digit.

Original entry on oeis.org

10123457689, 101723, 5437, 2339, 1009, 257, 139, 173, 83, 67, 31, 29, 37, 17, 17, 47, 19, 7, 5, 23, 23, 5, 11, 11, 17, 5, 5, 5, 5, 11, 5, 11, 11, 5, 5, 7, 5, 7, 3, 5, 5, 7, 7, 7, 3, 7, 3, 3, 5, 5, 5, 5, 3, 7, 7, 5, 3, 7, 5, 3, 3, 3, 3, 3, 3, 5, 3, 2, 3, 2, 3, 3, 3, 3, 5, 3, 3, 3, 2, 3, 5, 2

Offset: 1

Tanya Khovanova, Dec 05 2005

Conjecture: a(n)=2 for all n>168. Checked up to n = 20000. - Robert Israel, Aug 28 2020

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

Cf. A020666, A020665, A007377, A137214.

f:= proc(n) local k;
  k:= 1:
  do k:= nextprime(k);
    if convert(convert(k^n,base,10),set) = {$0..9} then return k fi
  od
end proc:
f(1):= 10123457689:
map(f, [$1..100]); # Robert Israel, Aug 28 2020

f[n_] := Block[{k = 1}, While[ Union@IntegerDigits[ Prime[k]^n] != {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, k++ ]; Prime[k]]; Array[f, 82] (* Robert G. Wilson v, Dec 06 2005 *)

from sympy import nextprime
def a(n):
    if n == 1: return 10123457689
    p = 2
    while not(len(set(str(p**n))) == 10): p = nextprime(p)
    return p
print([a(n) for n in range(1, 83)]) # Michael S. Branicky, Jul 04 2021

More terms from Robert G. Wilson v, Dec 06 2005

A131625 Numbers k such that decimal expansion of 3^k contains no 2.

Original entry on oeis.org

0, 1, 2, 4, 8, 9, 10, 11, 12, 15, 20, 22, 29, 34, 35, 54, 59

Offset: 1

Shyam Sunder Gupta, Sep 01 2007

I conjecture that 59 is the last term.

Cf. similar sequences listed in A131613.

Cf. A007377.

[n: n in [0..1000] | not 2 in Intseq(3^n)]; // Vincenzo Librandi, May 06 2015

Join[{0}, Select[ Range@10000, FreeQ[ IntegerDigits[3^# ], 2] &]]

isok(n) = ! vecsearch(Set(digits(3^n)), 2); \\ Michel Marcus, Feb 09 2018

Initial 0 added and Mathematica code adapted by Vincenzo Librandi, May 06 2015

A195985 Least prime such that p^2 is a zeroless n-digit number.

Original entry on oeis.org

2, 5, 11, 37, 107, 337, 1061, 3343, 10559, 33343, 105517, 333337, 1054133, 3333373, 10540931, 33333359, 105409309, 333333361, 1054092869, 3333333413, 10540925639, 33333333343, 105409255363, 333333333367, 1054092553583, 3333333333383, 10540925534207

Offset: 1

M. F. Hasler, Sep 26 2011

			a(1)^2=4, a(2)^2=25, a(3)^2=121, a(4)^2=1369 are the least squares of primes with 1, 2, 3 resp. 4 digits, and these digits are all nonzero.
a(5)=107 since 101^2=10201 and 103^2=10609 both contain a zero digit, but 107^2=11449 does not.
a(1000)=[10^500/3]+10210 (500 digits), since primes below sqrt(10^999) = 10^499*sqrt(10) ~ 3.162e499 have squares of less than 1000 digits, between sqrt(10^999) and 10^500/3 = sqrt(10^1000/9) ~ 3.333...e499 they have at least one zero digit. Finally, the 7 primes between 10^500/3 and a(1000) also happen to have a "0" digit in their square, but not so
  a(1000)^2 = 11111...11111791755555...55555659792849
  = [10^500/9]*(10^500+5) + 6806*10^500+104237294.

M. F. Hasler, Table of n, a(n) for n = 1..158
C. Rivera, Puzzle 607. A zeroless Prime power, on primepuzzles.net, Sept. 24, 2011.
W. Schneider, NoZeros: Powers n^k without Digit Zero (local copy of www.wschnei.de/digit-related-numbers/nozeros.html), as of Jan 30 2003.

Cf. A195942, A195943, A195944, A195945, A195946, A195908, A195948, A007377, A008839, A030700, A030701, A030702, A030703, A030704, A030705, A030706.

a(n)={ my(p=sqrtint(10^n\9)-1); until( is_A052382(p^2), p=nextprime(p+2));p}

cp's OEIS Frontend

A305942 Number of powers of 2 having exactly n digits '0' (in base 10), 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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A131613 Numbers k such that the decimal expansion of 3^k contains no 9.

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A130696 Numbers k such that 2^k does not contain all ten decimal digits.

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A050723 Numbers k such that the decimal expansion of 2^k contains no pair of consecutive equal digits (probably finite).

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A131629 Numbers k such that the decimal expansion of 3^k contains no 3.

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

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A112388 a(n) is the smallest prime p such that p^n contains every digit.

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