A030703 -id:A030703 - cp's OEIS frontend

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

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

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.

A238938 Powers of 2 without the digit '0' in their decimal expansion.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 8192, 16384, 32768, 65536, 262144, 524288, 16777216, 33554432, 134217728, 268435456, 2147483648, 4294967296, 8589934592, 17179869184, 34359738368, 68719476736, 137438953472, 549755813888, 562949953421312, 2251799813685248, 147573952589676412928

Offset: 1

M. F. Hasler, Mar 07 2014

Conjectured to be finite and complete. See the OEIS wiki page for further information, references and links.

			256 = 2^8 is in the sequence because 256 has a 2, a 5 and a 6 but no 0's.
512 = 2^9 is also in because it has a 1, a 2 and a 5 but no 0's.
1024 = 2^10 is not in the sequence because it has a 0.

Daniel Mondot, Table of n, a(n) for n = 1..36
M. F. Hasler, Zeroless powers, OEIS wiki, Mar 07 2014

Cf. A007377, A030700, A030701, A008839, A030702, A030703, A030704, A030705, A030706, A195944.
Cf. A195908, A195946, A195945.
Cf. A195942, A195943, A027870, A102483.

Select[2^Range[0, 127], DigitCount[#, 10, 0] == 0 &] (* Alonso del Arte, Mar 07 2014 *)

for(n=0,99,vecmin(digits(2^n))&& print1(2^n","))

a(n) = 2^A007377(n).

'fini' keyword removed as finiteness is only conjectured by Max Alekseyev, Apr 10 2019

A238939 Powers of 3 without the digit '0' in their decimal expansion.

Original entry on oeis.org

1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 177147, 531441, 1594323, 4782969, 1162261467, 94143178827, 282429536481, 2541865828329, 7625597484987, 22876792454961, 617673396283947, 16677181699666569, 278128389443693511257285776231761

Offset: 1

M. F. Hasler, Mar 07 2014

Conjectured to be finite and complete. See the OEIS wiki page for further information, references and links.

M. F. Hasler, Zeroless powers, OEIS wiki, Mar 07 2014.

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

Select[3^Range[0,100],DigitCount[#,10,0]==0&] (* Paolo Xausa, Oct 07 2023 *)

for(n=0,99,vecmin(digits(3^n))&& print1(3^n","))

a(n) = 3^A030700(n).

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

A238936 Powers of 6 without the digit '0' in their decimal expansion.

Original entry on oeis.org

1, 6, 36, 216, 1296, 7776, 46656, 279936, 1679616, 2176782336, 16926659444736, 4738381338321616896, 36845653286788892983296, 17324272922341479351919144385642496

Offset: 1

M. F. Hasler, Mar 07 2014

Conjectured to be finite and complete. See the OEIS wiki page for further information, references and links.

M. F. Hasler, Zeroless powers, OEIS wiki, Mar 07 2014

Cf. A007377, A030700, A030701, A008839, A030702, A030703, A030705, A030706, A195944.
Cf. A195908, A195946, A195945.
Cf. A195942, A195943, A027870, A102483.

Select[6^Range[0,50],DigitCount[#,10,0]==0&] (* Harvey P. Dale, Dec 03 2020 *)

for(n=0,99,vecmin(digits(6^n))&& print1(6^n","))

a(n)=6^A030702(n).

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

A238940 Powers of 4 without the digit '0' in their decimal expansion.

Original entry on oeis.org

1, 4, 16, 64, 256, 16384, 65536, 262144, 16777216, 268435456, 4294967296, 17179869184, 68719476736, 4722366482869645213696, 75557863725914323419136, 77371252455336267181195264

Offset: 1

M. F. Hasler, Mar 07 2014

Conjectured to be finite and complete. See the OEIS wiki page for further information, references and links.

M. F. Hasler, Zeroless powers, OEIS Wiki, Mar 07 2014

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

Select[4^Range[0,50],DigitCount[#,10,0]==0&] (* Harvey P. Dale, Aug 31 2021 *)

for(n=0,99,vecmin(digits(4^n))&& print1(4^n","))

a(n)=4^A030701(n).

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

A071531 Smallest exponent r such that n^r contains at least one zero digit (in base 10).

Original entry on oeis.org

10, 10, 5, 8, 9, 4, 4, 5, 1, 5, 4, 6, 7, 4, 3, 7, 4, 4, 1, 5, 3, 6, 6, 4, 6, 5, 5, 4, 1, 6, 2, 2, 3, 4, 5, 3, 4, 5, 1, 5, 3, 3, 4, 2, 5, 2, 2, 2, 1, 2, 2, 2, 4, 2, 5, 4, 6, 3, 1, 5, 6, 3, 2, 4, 6, 3, 9, 3, 1, 2, 6, 3, 3, 4, 8, 4, 2, 3, 1, 4, 5, 5, 2, 4, 3, 3, 6, 3, 1, 5, 5, 3, 3, 2, 7, 2, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 2

Paul Stoeber (paul.stoeber(AT)stud.tu-ilmenau.de), Jun 02 2002

For all n, a(n) is at most 40000, as shown below. Is 10 an upper bound?
If n has d digits, the numbers n, n^2, ..., n^k have a total of about N = k*(k+1)*d/2, and if these were chosen randomly the probability of having no zeros would be (9/10)^N. The expected number of d-digit numbers n with f(n)>k would be 9*10^(d-1)*(9/10)^N. If k >= 7, (9/10)^(k*(k+1)/2)*10 < 1 so we would expect heuristically that there should be only finitely many n with f(n) > 7. - Robert Israel, Jan 15 2015
The similar definition using "...exactly one digit 0..." would be ill-defined for all multiples of 100 and others (1001, ...). - M. F. Hasler, Jun 25 2018
When r=40000, one of the last five digits of n^r is always 0. Working modulo 10^5, we have 2^r=9736 and 5^r=90625, and both of these are idempotent; also, if gcd(n,10)=1, then n^r=1, and if 10|n, then n^r=0. Therefore the last five digits of n^r are always either 00000, 00001, 09736, or 90625. In particular, a(n) <= 40000. - Mikhail Lavrov, Nov 18 2021

			a(4)=5 because 4^1=4, 4^2=16, 4^3=64, 4^4=256, 4^5=1024 (has zero digit).

Robert Israel, Table of n, a(n) for n = 2..10000
OEIS Wiki, Zeroless powers (2014).

Cf. A305941 for the actual powers n^k.
Cf. A007377, A030700, A030701, A008839, A030702, A030703, A030704, A030705, A030706, A195944: decimal expansion of k^n contains no zeros, k = 2, 3, 4, ...
Cf. A305932, A305933, A305924, ..., A305929: row n = {k: x^k has n 0's}, x = 2, 3, ..., 9.
Cf. A305942, ..., A305947, A305938, A305939: #{k: x^k has n 0's}, x = 2, 3, ..., 9.
Cf. A306112, ..., A306119: largest k: x^k has n 0's; x = 2, 3, ..., 9.

f:= proc(n) local j;
for j from 1 do if has(convert(n^j,base,10),0) then return j fi od:
end proc:
seq(f(n),n=2..100); # Robert Israel, Jan 15 2015

zd[n_]:=Module[{r=1},While[DigitCount[n^r,10,0]==0,r++];r]; Array[zd,110,2] (* Harvey P. Dale, Apr 15 2012 *)

A071531(n)=for(k=1, oo, vecmin(digits(n^k))||return(k)) \\ M. F. Hasler, Jun 23 2018

def a(n):
    r, p = 1, n
    while 1:
        if "0" in str(p):
            return r
        r += 1
        p *= n
[a(n) for n in range(2, 100)] # Tim Peters, May 19 2005

a(n) >= 1 with equality iff n is in A011540 \ {0} = {10, 20, ..., 100, 101, ...}. - M. F. Hasler, Jun 23 2018

A305947 Number of powers of 7 having exactly n digits '0' (in base 10), conjectured.

Original entry on oeis.org

10, 11, 12, 13, 9, 10, 9, 7, 10, 14, 21, 10, 18, 7, 11, 11, 12, 15, 17, 10, 11, 6, 10, 16, 13, 9, 7, 9, 11, 12, 10, 16, 7, 16, 9, 14, 13, 13, 9, 17, 14, 12, 11, 9, 13, 9, 12, 12, 9, 12, 14

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) = 10 is the number of terms in A030703 and in A195908, which includes the power 7^0 = 1.

These are the row lengths of A305927. 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.

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. A030703 (= row 0 of A305927): k such that 7^k has no 0's; A195908: these powers 7^k.
Cf. A020665: largest k such that n^k has no '0's.
Cf. A063606 (= column 1 of A305927): least k such that 7^k has n digits '0' in base 10.
Cf. A305942 (analog for 2^k), ..., A305946, A305938, A305939 (analog for 9^k).

A305947(n,M=99*n+199)=sum(k=0,M,#select(d->!d,digits(7^k))==n)

A305947_vec(nMax,M=99*nMax+199,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(7^k)),nMax)]++);a[^-1]}

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

cp's OEIS Frontend

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

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A238938 Powers of 2 without the digit '0' in their decimal expansion.

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A238939 Powers of 3 without the digit '0' in their decimal expansion.

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A238936 Powers of 6 without the digit '0' in their decimal expansion.

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A238940 Powers of 4 without the digit '0' in their decimal expansion.

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