A031142 -id:A031142 - cp's OEIS frontend

A031143 Position of rightmost 0 (including leading 0) in 2^A031142(n).

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 21, 22, 23, 24, 25, 26, 36, 38, 54, 57, 59, 93, 115, 119, 120, 121, 136, 138, 164, 174, 176, 191, 196, 212, 217, 227, 233, 249, 250, 260, 268, 275, 308

Offset: 1

Matthew Cook, Dan Hoey, Eric W. Weisstein, David W. Wilson

nonn

"Positions" are counted 0,1,2,3,... starting with the least significant digit.

Cf. A031140, A031141, A031142, A031143.

best = 0; lst = {};
x = Select[Range[0, 10000],
  If[(t = First@
        First@StringPosition[StringReverse@("0" <> ToString@(2^#)),
          "0"]) > best, best = t; AppendTo[lst, t - 1]; True] &] ; lst (* Robert Price, Oct 11 2019 *)

More terms from Dan Hoey
a(42)-a(44) from Alan Griffiths, Jan 25 2012
a(45) from Alan Griffiths, Feb 01 2012
a(46) from Alan Griffiths, Mar 09 2012

A031156 a(n) = 2^A031142(n).

Original entry on oeis.org

1, 16, 128, 8192, 16384, 262144, 16777216, 134217728, 2147483648, 17179869184, 137438953472, 562949953421312, 2251799813685248, 147573952589676412928, 4722366482869645213696, 75557863725914323419136, 151115727451828646838272, 2417851639229258349412352, 77371252455336267181195264, 680564733841876926926749214863536422912

Offset: 1

N. J. A. Sloane

Powers of 2 such that there is an increasing number of nonzero digits after the rightmost digit 0, including a leading 0. a(20)=2^129 is the first term where the rightmost 0 is not the leading 0. - Hugo Pfoertner, Apr 12 2020

Cf. A031142, A031155.

A007377 Numbers k such that the decimal expansion of 2^k contains no 0.

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

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

It is an open problem of long standing to show that 86 is the last term.
A027870(a(n)) = A224782(a(n)) = 0. - Reinhard Zumkeller, Apr 30 2013
See A030700 for the analog for 3^k, which seems to end with k=68. - M. F. Hasler, Mar 07 2014
Checked up to k = 10^10. - David Radcliffe, Aug 21 2022

			Here is 2^86, conjecturally the largest power of 2 not containing a 0: 77371252455336267181195264. - _N. J. A. Sloane_, Feb 10 2023

J. S. Madachy, Mathematics on Vacation, Scribner's, NY, 1966, p. 126.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

W. Schneider, NoZeros: Powers n^k without Digit Zero [Cached copy]
Eric Weisstein's World of Mathematics, Zero
R. G. Wilson, V, Letter to N. J. A. Sloane, Oct. 1993

Cf. A027870, A030700, A102483, A034293.
Some similar sequences are listed in A035064.
Cf. also A031142.

import Data.List (elemIndices)
a007377 n = a007377_list !! (n-1)
a007377_list = elemIndices 0 a027870_list
-- Reinhard Zumkeller, Apr 30 2013

[ n: n in [0..50000] | not 0 in Intseq(2^n) ];  // Bruno Berselli, Jun 08 2011

remove(t -> has(convert(2^t,base,10),0),[$0..1000]); # Robert Israel, Dec 29 2015

Do[ If[ Union[ RealDigits[ 2^n ] [[1]]] [[1]] != 0, Print[ n ] ], {n, 1, 60000}]
Select[Range@1000, First@Union@IntegerDigits[2^# ] != 0 &]
Select[Range[0,100],DigitCount[2^#,10,0]==0&] (* Harvey P. Dale, Feb 06 2015 *)

for(n=0,99,if(vecmin(eval(Vec(Str(2^n)))),print1(n", "))) \\ Charles R Greathouse IV, Jun 30 2011

use bignum;
for(0..99) {
  if((1<<$_) =~ /^[1-9]+$/) {
    print "$_, "
  }
} # Charles R Greathouse IV, Jun 30 2011

def ok(n): return '0' not in str(2**n)
print(list(filter(ok, range(10**4)))) # Michael S. Branicky, Aug 08 2021

a(1) = 0 prepended by Reinhard Zumkeller, Apr 30 2013

A239008 Exponents m such that the decimal expansion of 3^m exhibits its first zero from the right later than any previous exponent.

Original entry on oeis.org

0, 3, 5, 7, 9, 11, 13, 19, 23, 24, 26, 28, 31, 34, 52, 65, 68, 136, 237, 4947, 7648, 42073, 50693, 52728, 395128, 2544983, 6013333, 76350564, 160451107, 641814146, 5291528429, 5856442430, 7307126644, 11577159988, 51444010646, 60457925746

Offset: 1

Charles R Greathouse IV and Robert G. Wilson v, Mar 14 2014

Assume that a zero precedes all decimal expansions. This will take care of those cases in A030700.
Inspired by the Seqfan list discussion Re: "possible sequence", beginning with David Wilson 7:57 PM Mar 06 2014 and continued by M. F. Hasler, Allan Wechsler and Franklin T. Adams-Watters.
Location of first zeros (from the right) of terms: 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 16, 18, 21, 22, 34, 57, 82, 84, 99, 103, 104, 139, 144, 151, 166, 169, 173, 202, 204, 205, 220, 230, 233, 236. - Chai Wah Wu, Jan 06 2020

			Obviously a(1) is 0. a(2) is 3 since this is the first exponent which yields a two-digit (nonzero) power of three.

Cf. A000244, A030700, A020665, A031142, A239009, A239010, A239011, A239012, A239013, A239014, A239015.

f[n_] := Position[ Reverse@ Join[{0}, IntegerDigits[ PowerMod[3, n, 10^500]]], 0, 1, 1][[1, 1]]; k = 1; mx = 0; lst = {}; While[k < 200000001, c = f[k]; If[c > mx, mx = c; AppendTo[ lst, k]; Print@ k]; k++]; lst

a(30)-a(34) from Bert Dobbelaere, Jan 21 2019

a(35)-a(36) from Chai Wah Wu, Jan 06 2020

A239009 Exponents m such that the decimal expansion of 4^m exhibits its first zero from the right later than any previous exponent.

Original entry on oeis.org

0, 2, 4, 7, 9, 12, 14, 16, 17, 23, 34, 36, 38, 43, 77, 88, 216, 350, 979, 24186, 28678, 134759, 205829, 374627, 2200364, 16625243, 29451854, 162613199, 8078176309, 9252290259, 17556077280, 49718535383, 51616746477, 54585993918

Offset: 1

Charles R Greathouse IV and Robert G. Wilson v, Mar 14 2014

Assume that a zero precedes all decimal expansions. This will take care of those cases in A030701.
Inspired by the seqfan list discussion Re: "possible sequence", beginning with David Wilson 7:57 PM Mar 06 2014 and continued by M. F. Hasler, Allan C. Wechsler and Franklin T. Adams-Watters.
Not just twice A031142, although {16625243, 29451854, 162613199, 9252290259, 51616746477, 54585993918, 146235898847, 1360645542292} are possible candidates.
Location of first zeros (from the right) of terms: 2, 3, 4, 6, 7, 9, 10, 11, 12, 13, 14, 23, 24, 27, 30, 39, 53, 58, 94, 113, 120, 121, 122, 139, 165, 177, 192, 213, 217, 228, 229, 230, 250, 251. - Chai Wah Wu, Jan 08 2020

Cf. A000302, A030701, A020665, A031142, A239008, A239010, A239011, A239012, A239013, A239014, A239015.

f[n_] := Position[ Reverse@ Join[{0}, IntegerDigits[ PowerMod[4, n, 10^500]]], 0, 1, 1][[1, 1]]; k = mx = 0; lst = {}; While[k < 100000001, c = f[k]; If[c > mx, mx = c; AppendTo[ lst, k]; Print@ k]; k++]; lst

a(28)-a(30) from Bert Dobbelaere, Jan 21 2019

a(31)-a(34) from Chai Wah Wu, Jan 08 2020

A239010 Exponents m such that the decimal expansion of 5^m exhibits its first zero from the right later than any previous exponent.

Original entry on oeis.org

0, 2, 3, 5, 6, 9, 11, 15, 17, 18, 25, 26, 30, 33, 57, 58, 153, 1839, 3290, 4081, 16431, 577839, 2190974, 15167023, 23155442, 24477994, 36290003, 53687441, 62497567, 181850218, 790111167, 872257561, 4531889178, 26964400609, 32626158305, 268600630073

Offset: 1

Charles R Greathouse IV and Robert G. Wilson v, Mar 14 2014

Assume that a zero precedes all decimal expansions. This will take care of those cases in A008839.
Inspired by the seqfan list discussion Re: "possible sequence", beginning with David Wilson 7:57 PM Mar 06 2014 and continued by M. F. Hasler, Allan C. Wechsler and Franklin T. Adams-Watters.
Highest position known is 232th digit from the right for a(33). - Bert Dobbelaere, Jan 21 2019

Cf. A000351, A008839, A020665, A031142, A239008, A239009, A239011, A239012, A239013, A239014, A239015.

f[n_] := Position[ Reverse@ Join[{0}, IntegerDigits[ PowerMod[5, n, 10^500]]], 0, 1, 1][[1, 1]]; k = mx = 0; lst = {}; While[k < 100000001, c = f[k]; If[c > mx, mx = c; AppendTo[ lst, k]; Print@ k]; k++]; lst

a(30)-a(33) from Bert Dobbelaere, Jan 21 2019

a(34)-a(36) from Chai Wah Wu, Jan 18 2020

A239011 Exponents m such that the decimal expansion of 6^m exhibits its first zero from the right later than any previous exponent.

Original entry on oeis.org

0, 2, 3, 4, 6, 7, 8, 12, 17, 24, 29, 42, 44, 101, 104, 128, 1015, 1108, 2629, 9683, 676076, 917474, 34882222, 53229360, 58230015, 90064345, 309000041, 319582553, 342860474, 382090917, 2770253437, 4380407969, 4407585753, 6966554399, 21235488251, 99404304146

Offset: 1

Charles R Greathouse IV and Robert G. Wilson v, Mar 14 2014

Assume that a zero precedes all decimal expansions. This will take care of those cases in A030702.

Inspired by the seqfan list discussion Re: "possible sequence", beginning with David Wilson 7:57 PM Mar 06 2014 and continued by M. F. Hasler, Allan C. Wechsler and Franklin T. Adams-Watters.

Cf. A000400, A030702, A020665, A031142, A239008, A239009, A239010, A239012, A239013, A239014, A239015.

f[n_] := Position[ Reverse@ Join[{0}, IntegerDigits[ PowerMod[6, n, 10^500]]], 0, 1, 1][[1, 1]]; k = mx = 0; lst = {}; While[k < 10000001, c = f[k]; If[c > mx, mx = c; AppendTo[ lst, k]; Print@ k]; k++]; lst

a(27)-a(34) from Bert Dobbelaere, Jan 21 2019

a(35)-a(36) from Chai Wah Wu, Jan 23 2020

A239012 Exponents m such that the decimal expansion of 7^m exhibits its first zero from the right later than any previous exponent.

Original entry on oeis.org

0, 2, 3, 6, 10, 11, 19, 35, 127, 131, 175, 207, 1235, 2470, 2651, 1241310, 1922910, 471056338, 1001431598, 1720335627, 4203146094, 5353516238, 21838571507, 25770284079, 40822793867

Offset: 1

Charles R Greathouse IV and Robert G. Wilson v, Mar 14 2014

Assume that a zero precedes all decimal expansions. This will take care of those cases in A030703.

Inspired by the seqfan list discussion Re: "possible sequence", beginning with David Wilson 7:57 PM Mar 06 2014 and continued by M. F. Hasler, Allan C. Wechsler and Franklin T. Adams-Watters.

Cf. A000420, A030703, A020665, A031142, A239008, A239009, A239010, A239011, A239013, A239014, A239015.

f[n_] := Position[ Reverse@ Join[{0}, IntegerDigits[ PowerMod[7, n, 10^500]]], 0, 1, 1][[1, 1]]; k = mx = 0; lst = {}; While[k < 500000001, c = f[k]; If[c > mx, mx = c; AppendTo[ lst, k]; Print@ k]; k++]; lst

a(19)-a(22) from Bert Dobbelaere, Jan 21 2019

a(23)-a(25) from Chai Wah Wu, Jan 15 2020

A239013 Exponents m such that the decimal expansion of 8^m exhibits its first zero from the right later than any previous exponent.

Original entry on oeis.org

0, 2, 3, 5, 6, 8, 9, 11, 12, 13, 17, 24, 27, 43, 144, 342, 633, 653, 2642, 6966, 16124, 84595, 225177, 4069057, 4890280, 6298187, 39573326, 99250579, 242281125, 1007075831, 4705063695, 5439666500, 5741331846, 6168193506, 9297912451, 34411164318, 36390662612, 265816303567

Offset: 1

Charles R Greathouse IV and Robert G. Wilson v, Mar 14 2014

Assume that a zero precedes all decimal expansions. This will take care of those cases in A030704.
Inspired by the seqfan list discussion Re: "possible sequence", beginning with David Wilson 7:57 PM Mar 06 2014 and continued by M. F. Hasler, Allan C. Wechsler and Franklin T. Adams-Watters.
Not just three times A031142; although {99250579, 6168193506, 9297912451, 34411164318, 36390662612} are possible candidates.

Cf. A001018, A030704, A020665, A031142, A239008, A239009, A239010, A239011, A239012, A239014, A239015.

f[n_] := Position[ Reverse@ Join[{0}, IntegerDigits[ PowerMod[8, n, 10^500]]], 0, 1, 1][[1, 1]]; k = mx = 0; lst = {}; While[k < 200000001, c = f[k]; If[c > mx, mx = c; AppendTo[ lst, k]; Print@ k]; k++]; lst

a(29)-a(35) from Bert Dobbelaere, Jan 21 2019

a(36)-a(38) from Chai Wah Wu, Jan 18 2020

A239014 Exponents m such that the decimal expansion of 9^m exhibits its first zero from the right later than any previous exponent.

Original entry on oeis.org

0, 2, 3, 4, 6, 7, 12, 13, 14, 17, 26, 34, 68, 406, 926, 2227, 3379, 3824, 26364, 197564, 9669757, 11470439, 15754533, 18945654, 25742286, 38175282, 237545304, 320907073, 2928221215, 3653563322, 5788579994, 25722005323, 30228962873, 137527721034, 217558664165, 523648850797

Offset: 1

Charles R Greathouse IV and Robert G. Wilson v, Mar 14 2014

Assume that a zero precedes all decimal expansions. This will take care of those cases in A030705.
Inspired by the seqfan list discussion Re: "possible sequence", beginning with David Wilson 7:57 PM Mar 06 2014 and continued by M. F. Hasler, Allan C. Wechsler and Franklin T. Adams-Watters.
Not just two time A001019.

Except for its second term, A030705 is a subsequence.

Cf. A001019, A020665, A031142, A239008, A239009, A239010, A239011, A239012, A239013, A239015.

f[n_] := Position[ Reverse@ Join[{0}, IntegerDigits[ PowerMod[9, n, 10^500]]], 0, 1, 1][[1, 1]]; k = mx = 0; lst = {}; While[k < 10000001, c = f[k]; If[c > mx, mx = c; AppendTo[ lst, k]; Print@ k]; k++]; lst

a(27)-a(31) from Bert Dobbelaere, Jan 21 2019

a(32)-a(36) from Chai Wah Wu, Jan 13 2020

cp's OEIS Frontend

A031143 Position of rightmost 0 (including leading 0) in 2^A031142(n).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A031156 a(n) = 2^A031142(n).

Views

Author

Keywords

Comments

Crossrefs

A007377 Numbers k such that the decimal expansion of 2^k contains no 0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Perl

Python

Extensions

A239008 Exponents m such that the decimal expansion of 3^m exhibits its first zero from the right later than any previous exponent.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A239009 Exponents m such that the decimal expansion of 4^m exhibits its first zero from the right later than any previous exponent.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A239010 Exponents m such that the decimal expansion of 5^m exhibits its first zero from the right later than any previous exponent.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A239011 Exponents m such that the decimal expansion of 6^m exhibits its first zero from the right later than any previous exponent.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A239012 Exponents m such that the decimal expansion of 7^m exhibits its first zero from the right later than any previous exponent.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions