A020665 -id:A020665 - cp's OEIS frontend

A306119 Largest k such that 9^k has exactly n digits 0 (in base 10), conjectured.

34, 36, 68, 56, 65, 106, 144, 134, 119, 138, 154, 186, 194, 191, 219, 208, 247, 267, 199, 314, 292, 263, 319, 303, 307, 345, 431, 401, 375, 388, 413, 498, 488, 504, 465, 513, 565, 464, 481, 541, 568, 532, 588, 542, 600, 677, 649, 633, 613, 734, 627

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) is the largest term in A030705: exponents of powers of 9 without digit 0 in base 10.

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. A063626: least k such that 9^k has n digits 0 in base 10.
Cf. A305939: number of k's such that 9^k has n digits 0.
Cf. A305929: row n lists exponents of 9^k with n digits 0.
Cf. A030705: { k | 9^k has no digit 0 } : row 0 of the above.
Cf. A020665: largest k such that n^k has no digit 0 in base 10.
Cf. A071531: least k such that n^k contains a digit 0 in base 10.
Cf. A103663: least x such that x^n has no digit 0 in base 10.
Cf. A306112, ..., A306118: analog for 2^k, ..., 8^k.

A306119_vec(nMax,M=99*nMax+199,x=9,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(x^k)),nMax)]=k);a[^-1]}

Data corrected thanks to a remark by R. J. Mathar, by M. F. Hasler, Feb 11 2023

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

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

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

A090493 Least k such that n^k contains all the digits from 0 through 9, or 0 if no such k exists.

Original entry on oeis.org

0, 68, 39, 34, 19, 20, 18, 28, 24, 0, 23, 22, 22, 21, 12, 17, 14, 21, 17, 51, 17, 18, 14, 19, 11, 18, 13, 11, 12, 39, 11, 14, 16, 14, 19, 10, 13, 14, 17, 34, 11, 17, 13, 16, 15, 11, 12, 12, 9, 18, 16, 11, 13, 10, 12, 7, 13, 11, 11, 20, 14, 18, 13, 14, 10, 13, 10, 9, 11, 18, 15

Offset: 1

Amarnath Murthy, Dec 03 2003

Note that the values of n for which a(n) = 1 have density 1.
Is it known that a(n)=0 only for n a power of 10? - Christopher J. Smyth, Aug 21 2014
a(n) >= ceiling(log_n(10)*9), whenever a(n)>0. This is because in order for an integer to have 10 digits its base-10 magnitude must be at least 9. - Ely Golden, Sep 06 2017

			a(5)=19: 5^19 = 19073486328125.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Ely Golden, Python program to generate a(n)

Cf. A020665, A062518, A171102.

Exponents of powers of k that contain all ten decimal digits: A130694 (k=2), A236673 (k=3), A284670 (k=5), A284672 (k=7).

a:= proc(n) local k;
   if n = 10^ilog10(n) then return 0 fi;
   for k from 1 do
     if nops(convert(convert(n^k,base,10),set))=10 then return k fi
   od
end proc:
seq(a(n),n=1..100); # Robert Israel, Aug 20 2014

Table[If[IntegerQ@ Log10[n], 0, SelectFirst[Range[#, # + 100] &@ Ceiling[9 Log[n, 10]], NoneTrue[DigitCount[n^#], # == 0 &] &]], {n, 71}] (* Michael De Vlieger, Sep 06 2017 *)

a(n) = if (n == 10^valuation(n, 10), return (0)); k=1; while(#vecsort(digits(n^k),,8)!=10, k++); k; \\ Michel Marcus, Aug 20 2014

def a(n):
  s = str(n)
  if n == 1 or (s.count('0')==len(s)-1 and s.startswith('1')):
    return 0
  k = 1
  count = 0
  while count != 10:
    count = 0
    for i in range(10):
      if str(n**k).count(str(i)) == 0:
        count += 1
        break
    if count:
      k += 1
    else:
      return k
n = 1
while n < 100:
  print(a(n), end=', ')
  n += 1
# Derek Orr, Aug 20 2014

a(10^e) = 0; a(m^e) = a(m)/e for e dividing a(m). - Reinhard Zumkeller, Dec 06 2004

More terms from Reinhard Zumkeller, Dec 06 2004

Corrected a(15), a(17), a(38), a(48), a(56) and a(65). (For each of these terms, the only 1 in n^k is the first digit.) - Jon E. Schoenfield, Sep 20 2008

cp's OEIS Frontend

A306119 Largest k such that 9^k has exactly n digits 0 (in base 10), conjectured.

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

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

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

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

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

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

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

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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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