A031142 -id:A031142 - cp's OEIS frontend

A239015 Exponents m such that the decimal expansion of 11^m exhibits its first zero from the right later than any previous exponent.

0, 1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 18, 36, 41, 366, 488, 4357, 69137, 89371, 143907, 542116, 2431369, 5877361, 8966861, 121915452, 123793821, 221788016, 709455085, 1571200127, 2640630712, 6637360862, 64994336645, 74770246842

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 A001020.

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.

			Illustration of initial term, with the 0 enclosed in parentheses:
n, position of 0, 11^a(n)
1, 2, (0)1
2, 3, (0)11
3, 4, (0)121
4, 5, (0)1331
5, 6, (0)14641
6, 7, (0)1771561
7, 8, (0)19487171
8, 9, (0)214358881
9, 10, (0)2357947691
10, 11, (0)3138428376721
11, 12, (0)34522712143931
12, 13, (0)379749833583241
13, 14, (0)4177248169415651
14, 15, (0)45949729863572161
15, 16, (0)5559917313492231481
16, 17, 3091268053287(0)672635673352936887453361
...
- _N. J. A. Sloane_, Jan 16 2020

Cf. A001020, A030706, A020665, A031142, A239008, A239009, A239010, A239011, A239012, A239013, A239014.

f[n_] := Position[ Reverse@ Join[{0}, IntegerDigits[ PowerMod[11, n, 10^500]]], 0, 1, 1][[1, 1]]; k = mx = 0; lst = {}; While[k < 40000001, c = f[k]; If[c > mx, mx = c; AppendTo[ lst, k]; Print@ k]; k++]; lst

a(28)-a(34) from Bert Dobbelaere, Jan 22 2019

a(35)-a(36) from Chai Wah Wu, Jan 16 2020

A031140 Position of rightmost 0 in 2^n increases.

Original entry on oeis.org

10, 20, 30, 40, 46, 68, 93, 95, 129, 176, 229, 700, 1757, 1958, 7931, 57356, 269518, 411658, 675531, 749254, 4400728, 18894561, 33250486, 58903708, 297751737, 325226398, 781717865, 18504580518, 27893737353, 103233492954

Offset: 1

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

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

I.e., look for increasing number of nonzero digits after the rightmost digit '0'. - M. F. Hasler, Jun 21 2018

			From _M. F. Hasler_, Jun 21 2018: (Start)
2^10 = 1024 is the first power of 2 to have a digit '0', which is the third digit from the right, i.e., it has to its right no digit '0' and two nonzero digits.
2^20 = 1048576 is the next larger power with a digit '0' having to its right no digit '0' and more (namely 5) nonzero digits than the above 1024.
After 2^46 = 70368744177664 there is 2^52 = 4503599627370496 having a '0' further to the left, but this digit has another '0' to its right and therefore cannot be considered: The next term having more nonzero digits after its rightmost '0' is only 2^68. (End)

Eric Weisstein's World of Mathematics, Zero.

Cf. A031141, A031142, A031143.

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

m=0;for(k=0,oo,d=digits(2^k);for(j=0,#d-1,d[#d-j]||(j>m&&(m=j)&&print1(k",")||break))) \\ M. F. Hasler, Jun 21 2018

More terms from Dan Hoey.

A031141 Position of rightmost digit 0 in 2^A031140(n).

Original entry on oeis.org

2, 5, 8, 11, 12, 13, 14, 23, 36, 38, 54, 57, 59, 93, 115, 119, 120, 121, 136, 138, 164, 174, 176, 191, 196, 212, 217, 227, 233, 249

Offset: 1

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

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

Eric Weisstein's World of Mathematics, Zero.

Cf. A031140, A031142, A031143.

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

m=0;for(k=0,oo,d=digits(2^k);for(j=0,#d-1,d[#d-j]||(j>m&&print1(m=j,",")||break))) \\ M. F. Hasler, Jun 21 2018

More terms from Dan Hoey

A181611 Position of rightmost zero in 2^n (including leading zero).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 2, 2, 2, 4, 5, 5, 5, 2, 6, 6, 5, 5, 1, 1, 8, 8, 4, 9, 9, 3, 8, 10, 10, 10, 11, 11, 11, 12, 4, 12, 11, 8, 1, 1, 5, 5, 12, 12, 3, 15, 7, 16, 3, 3, 7, 8, 8, 8, 12, 7, 7, 10, 1, 1, 7, 4, 4, 21, 13, 7, 4, 4, 22, 6, 6, 4, 23, 24, 13, 2, 4, 25, 1, 1, 11, 6, 26, 3, 2, 12, 12, 12, 11, 14, 14, 23, 3, 3, 4, 4, 4, 3, 1, 1, 2, 2, 2, 6, 6, 8, 2, 2, 2, 3, 3, 3, 17, 2, 5, 6, 6, 6

Offset: 1

Tanya Khovanova, Jan 30 2011

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

			2^10 = 1024, the rightmost zero is in position 2, so a(10) = 2. Another example, 2^5 = 32, so we need to add a leading zero: 032, thus the rightmost zero will be in position 2, and a(5) = 2.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A031142, A031141.

Cf. A000079, A215887.

A181611 := proc(n) local dgs,i ; dgs := convert(2^n, base, 10) ; i := ListTools[Search](0, dgs) ; if i > 0 then i-1; else nops(dgs) ; end if ; end proc: # R. J. Mathar, Jan 30 2011
a:= proc(n) local m, i;
      m:= 2^n;
      for i from 0 while m>0 and irem(m, 10, 'm')<>0
      do od; i
    end:
seq(a(n), n=1..121);  # Alois P. Heinz, Feb 05 2011

Table[Position[Reverse[Prepend[IntegerDigits[2^n], 0]],
    0][[1]][[1]] - 1, {n, 121}]

a(n) = {my(d = Vecrev(digits(2^n))); for (i=1, #d, if (!d[i], return (i-1));); #d;} \\ Michel Marcus, Jan 01 2016

a(n) = A215887(A000079(n)). - Michel Marcus, Jan 01 2016

A266568 a(n) = smallest k such that 2^k ends in a string of exactly n nonzero digits.

Original entry on oeis.org

0, 4, 7, 13, 14, 18, 50, 24, 27, 31, 34, 37, 68, 93, 49, 51, 116, 214, 131, 155, 67, 72, 76, 77, 81, 86, 149, 498, 154, 286, 359, 866, 1225, 329, 664, 129, 573, 176, 655, 820, 571, 434, 1380, 475, 1260, 2251, 6015, 3066, 1738, 2136, 2297, 432, 665, 229, 1899

Offset: 1

Jon E. Schoenfield, Jan 01 2016

Since 2^a(n) must have at least n digits, a(n) >= (n-1)*log_2(10).
The 26-digit number 2^86 = 77371252455336267181195264 is almost certainly the largest power of 2 that contains no zero digit.
A notably low local minimum occurs at a(36) = 129, which is less than a(n) for all n > 26.
A notably high local maximum occurs at a(122) = 11267047.

			2^0 = 1 is the smallest power of 2 ending in a string ("1") of exactly 1 nonzero digit, so a(1) = 0.
2^4 = 16 is the smallest power of 2 ending in a string ("16") of exactly 2 nonzero digits, so a(2) = 4.
2^50 = 1125899906842624 is the smallest power of 2 ending in a string ("6842624") of exactly 7 nonzero digits, so a(7) = 50.
The last 7 digits of 2^24 = 16777216 -- i.e., "6777216" -- are also nonzero, but so is the preceding digit, so 2^24 ends in a string of exactly 8 nonzero digits. Since no smaller power of 2 ends in exactly 8 nonzero digits, a(8) = 24.

Cf. A007377, A031140, A031141, A031142, A031143, A181611.

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A239015 Exponents m such that the decimal expansion of 11^m exhibits its first zero from the right later than any previous exponent.

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A031140 Position of rightmost 0 in 2^n increases.

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A031141 Position of rightmost digit 0 in 2^A031140(n).

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A181611 Position of rightmost zero in 2^n (including leading zero).

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A266568 a(n) = smallest k such that 2^k ends in a string of exactly n nonzero digits.

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