A007377 -id:A007377 - cp's OEIS frontend

A030704 Numbers k such that the decimal expansion of 8^k contains no zeros (probably finite).

0, 1, 2, 3, 5, 6, 8, 9, 11, 12, 13, 17, 24, 27

Offset: 1

Integers in A007377 / 3. - M. F. Hasler, Mar 07 2014

Eric Weisstein's World of Mathematics, Zero

Cf. A007377 (analog for 2^n), A030700 (for 3^n), A030701 (for 4^n), A008839 (for 5^n), A030702 (for 6^n), A030703 and A195908 (for 7^n), A030705 (for 9^n), A030706 and A195946 (for 11^n), A195944 and A195945 (for 13^n).
Cf. A195942, A195943.
This is row 0 of A305928.

[n: n in [0..500] | not 0 in Intseq(8^n)]; // Vincenzo Librandi, Mar 08 2014

Select[Range[0,30],DigitCount[8^#,10,0]==0&] (* Harvey P. Dale, Jul 13 2016 *)

select( is(n)=vecmin(digits(8^n)), [0..30]) \\ M. F. Hasler, Mar 07 2014

Several edits (offset 1, initial 0, title rephrased) by M. F. Hasler, Mar 07 2014

A195908 Powers of 7 which have no zero in their decimal expansion.

Original entry on oeis.org

1, 7, 49, 343, 117649, 823543, 282475249, 1977326743, 11398895185373143, 378818692265664781682717625943

Offset: 1

M. F. Hasler, Sep 25 2011

Probably finite. Is 378818692265664781682717625943 the largest term?

No further terms up to 7^50,000, a number with 42,255 digits. - Harvey P. Dale, Jul 14 2022

M. F. Hasler, Zeroless powers, OEIS Wiki, Mar 07 2014
C. Rivera, Puzzle 607. A zeroless Prime power, on primepuzzles.net, Sept. 24, 2011.
W. Schneider, NoZeros: Powers n^k without Digit Zero (local copy of www.wschnei.de/digit-related-numbers/nozeros.html), as of Jan 30 2003.

Cf. A195942, A195943, A195944, A195945, A195946, A195948, A007377, A008839, A030700, A030701, A030702, A030703, A030704, A030705, A030706.

[7^n: n in [0..3*10^4] | not 0 in Intseq(7^n)]; // Bruno Berselli, Sep 26 2011

Select[7^Range[0,50],DigitCount[#,10,0]==0&] (* Harvey P. Dale, Jul 14 2022 *)

for( n=1,9999, is_A052382(7^n) && print1(7^n,","))

a(n) = 7^A030703(n).

A000420 intersect A052382.

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

A195946 Powers of 11 which have no zero in their decimal expansion.

Original entry on oeis.org

1, 11, 121, 1331, 14641, 1771561, 19487171, 214358881, 2357947691, 3138428376721, 34522712143931, 379749833583241, 4177248169415651, 45949729863572161, 5559917313492231481, 4978518112499354698647829163838661251242411

Offset: 1

M. F. Hasler, Sep 25 2011

Probably finite. Is 4978518112499354698647829163838661251242411 the largest term?

M. F. Hasler, Zeroless powers, OEIS Wiki, Mar 07 2014
C. Rivera, Puzzle 607. A zeroless Prime power, on primepuzzles.net, Sept. 24, 2011.
W. Schneider, NoZeros: Powers n^k without Digit Zero (local copy of www.wschnei.de/digit-related-numbers/nozeros.html), as of Jan 30 2003.

For the zeroless numbers (powers x^n), see A195942, A195943, A238938, A238939, A238940, A195948, A238936, A195908, A195945.

For the corresponding exponents, see A007377, A008839, A030700, A030701, A030702, A030703, A030704, A030705, A030706, A195944.

[11^n: n in [0..3*10^4] | not 0 in Intseq(11^n)]; // Bruno Berselli, Sep 26 2011

Select[11^Range[0,50],DigitCount[#,10,0]==0&] (* Harvey P. Dale, Jan 27 2014 *)

for( n=0,9999, is_A052382(11^n) && print1(11^n,","))

a(n) = 11^A030706(n).

A195946 = A001020 intersect A052382.

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

A195943 Zeroless prime powers: Intersection of A000961 and A052382.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 251, 256, 257, 263, 269, 271, 277, 281, 283, 289, 293, 311

Offset: 1

M. F. Hasler, Sep 25 2011

In contrast to A195942, we also allow for primes (p^n with n=1) in this sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
C. Rivera, Puzzle 607. A zeroless Prime power, on primepuzzles.net, Sept. 24, 2011.

Cf. A195942, A195944, A195945, A195946, A195908, A195948, A007377, A008839, A030700, A030701, A030702, A030703, A030704, A030705, A030706.

Cf. A038618 (subsequence).

a195943 n = a195943_list !! (n-1)
a195943_list = filter ((== 1) . a010055) a052382_list
-- Reinhard Zumkeller, Sep 27 2011

for( n=1,9999, is_A000961(n) && is_A052382(n) && print1(n","))

A195943 = A000961 intersect A052382 = A000961 \ A011540.

A010055(a(n)) * A168046(a(n)) = 1. - Reinhard Zumkeller, Sep 27 2011

A195945 Powers of 13 which have no zero in their decimal expansion.

Original entry on oeis.org

1, 13, 169, 2197, 28561, 371293, 62748517, 137858491849, 3937376385699289

Offset: 1

M. F. Hasler, Sep 25 2011

Probably finite. Is 3937376385699289 the largest term?
No further terms up to 13^25000. - Harvey P. Dale, Oct 01 2011
No further terms up to 13^45000. - Vincenzo Librandi, Jul 31 2013
No further terms up to 13^(10^9). - Daniel Starodubtsev, Mar 22 2020

M. F. Hasler, Zeroless powers, OEIS Wiki, Mar 07 2014
C. Rivera, Puzzle 607. A zeroless Prime power, on primepuzzles.net, Sept. 24, 2011.
W. Schneider, NoZeros: Powers n^k without Digit Zero (local copy of www.wschnei.de/digit-related-numbers/nozeros.html), as of Jan 30 2003.

For other zeroless powers x^n, see A238938 (x=2), A238939, A238940, A195948, A238936, A195908, A195946 (x=11), A195945, A195942, A195943, A103662.
For the corresponding exponents, see A007377, A008839, A030700, A030701, A008839, A030702, A030703, A030704, A030705, A030706, A195944 and also A020665.
For other related sequences, see A052382, A027870, A102483, A103663.

[13^n: n in [0..2*10^4] | not 0 in Intseq(13^n)]; // Bruno Berselli, Sep 26 2011

Select[13^Range[0,250],DigitCount[#,10,0]==0&] (* Harvey P. Dale, Oct 01 2011 *)

for(n=0,9999, is_A052382(13^n) && print1(13^n,","))

Equals A001022 intersect A052382 (as a set).

Equals A001022 o A195944 (as a function).

A195942 Zeroless prime powers (excluding primes): Intersection of A025475 and A052382.

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1331, 1369, 1681, 1849, 2187, 2197, 3125, 3481, 3721, 4489, 4913, 5329, 6241, 6561, 6859, 6889, 7921, 8192

Offset: 1

M. F. Hasler, Sep 25 2011

Reinhard Zumkeller and T. D. Noe, Table of n, a(n) for n = 1..4094 (terms < 10^10)
C. Rivera, Puzzle 607. A zeroless Prime power, on primepuzzles.net, Sept. 24, 2011.

Cf. A195985, A195943, A195944, A195945, A195946, A195908, A195948, A007377, A008839, A030700, A030701, A030702, A030703, A030704, A030705, A030706.

a195942 n = a195942_list !! (n-1)
a195942_list = filter (\x -> a010051 x == 0 && a010055 x == 1) a052382_list
-- Reinhard Zumkeller, Sep 27 2011

mx = 10^10; t = {1}; p = 2; While[pw = 2; While[n = p^pw; n <= mx, If[Union[IntegerDigits[n]][[1]] > 0, AppendTo[t, n]]; pw++]; pw > 2, p = NextPrime[p]]; t = Sort[t] (* T. D. Noe, Sep 27 2011 *)

for( n=1,9999, is_A025475(n) && is_A052382(n) && print1(n","))

A195942 = A025475 intersect A052382.

A010055(a(n)) * (1 - A010051(a(n))) * A168046(a(n)) = 1. - Reinhard Zumkeller, Sep 27 2011

A195944 Numbers k such that 13^k has no zero in its decimal expansion.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 10, 14

Offset: 1

M. F. Hasler, Sep 25 2011

Probably finite. Is 14 the largest term?

Carlos Rivera, Puzzle 607. A zeroless Prime power

Cf. A195942, A195943, A195945, A195946, A195908, A195948, A052382, A007377, A008839, A030700, A030701, A030702, A030704, A030705, A030706.

[n: n in [0..1000] | not 0 in Intseq(13^n) ]; // Vincenzo Librandi, May 06 2015

Select[Range[0,20],DigitCount[13^#,10,0]==0&] (* Harvey P. Dale, May 24 2023 *)

for( n=0,9999, is_A052382(13^n) && print1(n","))

Equals { n | A001022(n) is in A052382 }.

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

A195948 Powers of 5 which have no zero in their decimal expansion.

Original entry on oeis.org

1, 5, 25, 125, 625, 3125, 15625, 78125, 1953125, 9765625, 48828125, 762939453125, 3814697265625, 931322574615478515625, 116415321826934814453125, 34694469519536141888238489627838134765625

Offset: 1

M. F. Hasler, Sep 25 2011

Probably finite. Is 34694469519536141888238489627838134765625 the largest term?

C. Rivera, Puzzle 607. A zeroless Prime power, on primepuzzles.net, Sept. 24, 2011.

Cf. A195943, A195944, A195945, A195946, A195908, A007377, A008839, A030700, A030701, A030702, A030703, A030704, A030705, A030706.

Select[5^Range[0,60],DigitCount[#,10,0]==0&] (* Harvey P. Dale, Aug 30 2016 *)

for( n=0,9999, is_A052382(5^n) && print1(5^n,","))

a(n) = 5^A008839(n).

A000351 intersect A052382.

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

A102483 Numbers k such that 2^k contains no zeros in base 3.

Original entry on oeis.org

0, 1, 2, 3, 4, 15

Offset: 1

N. J. A. Sloane, Feb 25 2005

I conjectured in 1973 that there are no further terms. This question is still open.
A104320(a(n)) = 0. - Reinhard Zumkeller, Mar 01 2005
No other terms less than 200000. - Robert G. Wilson v, Dec 06 2005
a(7) > 10^7. - Martin Ehrenstein, Jul 27 2021
If it exists, a(7) > 10^21. - Robert Saye, Mar 23 2022

Robert I. Saye, On two conjectures concerning the ternary digits of powers of two, J. Integer Seq. 25 (2022) Article 22.3.4.
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Ternary

Cf. A007377, A004642, A346497.

Select[ Range@1000, FreeQ[ IntegerDigits[2^#, 3], 0] &] (* Robert G. Wilson v, Dec 06 2005 *)

for (n=0, 100, if (vecmin(digits(2^n, 3)), print1(n, ", "))) \\ Michel Marcus, Mar 25 2015

A031146 Exponent of the least power of 2 having exactly n zeros in its decimal representation.

Original entry on oeis.org

0, 10, 42, 43, 79, 88, 100, 102, 189, 198, 242, 250, 252, 263, 305, 262, 370, 306, 368, 383, 447, 464, 496, 672, 466, 557, 630, 629, 628, 654, 657, 746, 771, 798, 908, 913, 917, 906, 905, 1012, 1113, 988, 1020, 989, 1044, 1114, 1120, 1118, 1221, 1218, 1255

Offset: 0

N. J. A. Sloane

			a(3) = 43 since 2^m contains 3 0's for m starting with 43 (2^43 = 8796093022208) and followed by 53, 61, 69, 70, 83, 87, 89, 90, 93, ...

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A000079, A007377, A031147, A006889.

Cf. A063555 (analog for 3^k), A063575 (for 4^k), A063585 (for 5^k), A063596 (for 6^k), A063606 (for 7^k), A063616 (for 8^k), A063626 (for 9^k).

a = {}; Do[k = 0; While[ Count[ IntegerDigits[2^k], 0] != n, k++ ]; a = Append[a, k], {n, 0, 50} ]; a (* Robert G. Wilson v, Jun 12 2004 *)
nn = 100; t = Table[0, {nn}]; found = 0; k = 0; While[found < nn, k++; cnt = Count[IntegerDigits[2^k], 0]; If[cnt <= nn && t[[cnt]] == 0, t[[cnt]] = k; found++]]; t = Join[{0}, t] (* T. D. Noe, Mar 14 2012 *)

A031146(n)=for(k=0, oo, #select(d->!d, digits(2^k))==n&&return(k)) \\ M. F. Hasler, Jun 15 2018

More terms from Erich Friedman

Definition clarified by Joerg Arndt, Sep 27 2016

cp's OEIS Frontend

A030704 Numbers k such that the decimal expansion of 8^k contains no zeros (probably finite).

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Magma

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A195908 Powers of 7 which have no zero in their decimal expansion.

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Magma

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A195946 Powers of 11 which have no zero in their decimal expansion.

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A195943 Zeroless prime powers: Intersection of A000961 and A052382.

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A195945 Powers of 13 which have no zero in their decimal expansion.

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A195942 Zeroless prime powers (excluding primes): Intersection of A025475 and A052382.

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A195944 Numbers k such that 13^k has no zero in its decimal expansion.

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