A238938 -id:A238938 - 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]}

A238985 Zeroless 7-smooth numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 14, 15, 16, 18, 21, 24, 25, 27, 28, 32, 35, 36, 42, 45, 48, 49, 54, 56, 63, 64, 72, 75, 81, 84, 96, 98, 112, 125, 126, 128, 135, 144, 147, 162, 168, 175, 189, 192, 196, 216, 224, 225, 243, 245, 252, 256, 288, 294, 315, 324, 336

Offset: 1

Charles R Greathouse IV and Reinhard Zumkeller, Mar 07 2014

A001221(a(n)) <= 3 since 10 cannot divide a(n).

It seems that this sequence is finite and contains 12615 terms. - Daniel Mondot, May 03 2022 and Jianing Song, Jan 28 2023

			a(12615) = 2^25 * 3^227 * 7^28.

Daniel Mondot, Table of n, a(n) for n = 1..12615 (terms 1..10000 from Charles R Greathouse IV)

Cf. A168046, intersection of A002473 and A052382.
A238938, A238939, A238940, A195948, A238936, A195908 are proper subsequences.
Cf. A059405 (subsequence), A350180 through A350187.

import Data.Set (singleton, deleteFindMin, fromList, union)
a238985 n = a238985_list !! (n-1)
a238985_list = filter ((== 1) . a168046) $ f $ singleton 1 where
   f s = x : f (s' `union` fromList
               (filter ((> 0) . (`mod` 10)) $ map (* x) [2,3,5,7]))
               where (x, s') = deleteFindMin s

zf(n)=vecmin(digits(n))
list(lim)=my(v=List(),t,t1); for(e=0,log(lim+1)\log(7), t1=7^e; for(f=0,log(lim\t1+1)\log(3), t=t1*3^f; while(t<=lim, if(zf(t), listput(v, t)); t<<=1)); for(f=0,log(lim\t1+1)\log(5), t=t1*5^f; while(t<=lim, if(zf(t), listput(v, t)); t*=3))); Set(v)

A086299(a(n)) * A168046(a(n)) = 1.

Keyword:fini and keyword:full removed by Jianing Song, Jan 28 2023 as finiteness is only conjectured.

A245853 Powers of 12 without the digit '0' in their decimal expansion.

Original entry on oeis.org

1, 12, 144, 1728, 248832, 2985984, 429981696, 61917364224, 1283918464548864, 3833759992447475122176, 11447545997288281555215581184

Offset: 1

Vincenzo Librandi, Aug 04 2014

Conjectured to be finite.

Cf. Powers of k without the digit '0' in their decimal expansion: A238938 (k=2), A238939 (k=3), A238940 (k=4), A195948 (k=5), A238936 (k=6), A195908 (k=7), A245852 (k=8), A240945 (k=9), A195946 (k=11), this sequence (k=12), A195945 (k=13).

[12^n: n in [0..3*10^4] | not 0 in Intseq(12^n)];

Select[12^Range[0, 2*10^5], DigitCount[#, 10, 0]==0 &]

A298607 Powers of 2 with the digit '0' in their decimal expansion.

Original entry on oeis.org

1024, 2048, 4096, 131072, 1048576, 2097152, 4194304, 8388608, 67108864, 536870912, 1073741824, 274877906944, 1099511627776, 2199023255552, 4398046511104, 8796093022208, 17592186044416, 35184372088832, 70368744177664, 140737488355328, 281474976710656, 1125899906842624

Offset: 1

Alonso del Arte, Jan 22 2018

The complement, A238938, is conjectured to be finite. Furthermore, Khovanova (see link) believes 2^86 = 77371252455336267181195264 is the largest power of 2 not in this sequence.

			2^12 = 4096 contains one 0 in its decimal representation, hence 4096 is in the sequence.
2^13 = 8192 contains no 0's and is thus not in the sequence.

Tanya Khovanova, "86 Conjecture", Tanya Khovanova's Math Blog, February 15, 2011.

Cf. A007377, A027870.

Select[2^Range[0, 63], DigitCount[#, 10, 0] > 0 &]

lista(nn) = {for (n=0, nn, if (vecsearch(Set(digits(p=2^n)), 0), print1(p, ", ")););} \\ Michel Marcus, Mar 05 2018

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, Dec 17 2014

Conjectured to be finite.

See A245853 for the actual powers 12^a(n).

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.
For other related sequences, see A052382, A027870, A102483, A103663.

Select[Range[0,30],DigitCount[12^#,10,0]==0&] (* Harvey P. Dale, Apr 06 2019 *)

for(n=0,9e9,vecmin(digits(12^n))&&print1(n","))

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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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A238985 Zeroless 7-smooth numbers.

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A245853 Powers of 12 without the digit '0' in their decimal expansion.

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A298607 Powers of 2 with the digit '0' in their decimal expansion.

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A252482 Exponents n such that the decimal expansion of the power 12^n contains no zeros.

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