A239015
Exponents m such that the decimal expansion of 11^m exhibits its first zero from the right later than any previous exponent.
Original entry on oeis.org
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
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.
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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
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
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.
Cf.
A195942,
A195943,
A195944,
A195945,
A195946,
A195908,
A195948,
A007377,
A008839,
A030700,
A030701,
A030702,
A030703,
A030704,
A030705,
A030706.
A272269
Numbers n such that 11^n 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, 25, 27, 28, 34, 38, 41
Offset: 1
25 is a term because 11^25 = 108347059433883722041830251 that does not contain digit 6.
26 is not a term because 11^26 = 11^25*10 + 11^25 = 1083470594338837220418302510 + 108347059433883722041830251 = 1191817653772720942460132761 that contains all ten decimal digits.
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Select[Range[0, 120], AnyTrue[DigitCount[11^#], # == 0 &] &] (* Michael De Vlieger, Apr 24 2016, Version 10 *)
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isA171102(n) = 9<#vecsort(Vecsmall(Str(n)), , 8);
lista(nn) = for(n=0, nn, if(!isA171102(11^n), print1(n, ", ")));
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select( is_A272269(n)=#Set(digits(11^n))<10 ,[0..100]) \\ M. F. Hasler, May 18 2017
A050731
Decimal expansion of 11^n contains no pair of consecutive equal digits (probably finite).
Original entry on oeis.org
0, 2, 4, 5, 7, 9, 10, 12, 16
Offset: 0
11^16 = 45949729863572161.
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, 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.
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Select[Range[0,30],DigitCount[12^#,10,0]==0&] (* Harvey P. Dale, Apr 06 2019 *)
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for(n=0,9e9,vecmin(digits(12^n))&&print1(n","))
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