A030700 -id:A030700 - cp's OEIS frontend

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

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.

A305933 Irregular table read by rows: row n >= 0 lists all k >= 0 such that the decimal representation of 3^k has n digits '0' (conjectured).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 19, 23, 24, 26, 27, 28, 31, 34, 68, 10, 15, 16, 17, 18, 20, 25, 29, 43, 47, 50, 52, 63, 72, 73, 22, 30, 32, 33, 36, 38, 39, 40, 41, 42, 44, 45, 46, 48, 51, 53, 56, 58, 60, 61, 62, 64, 69, 71, 83, 93, 96, 108, 111, 123, 136, 21, 37, 49, 67, 75, 81, 82, 87, 90, 105, 112, 121, 129

Offset: 0

M. F. Hasler, Jun 14 2018

The set of nonempty rows is a partition of the nonnegative integers.
Read as a flattened sequence, a permutation of the nonnegative integers.
In the same way, another choice of (basis, digit, base) = (m, d, b) different from (3, 0, 10) will yield a similar partition of the nonnegative integers, trivial if m is a multiple of b.
It remains an open problem to provide a proof that the rows are complete, just as each of the terms of A020665 is unproved.
We can also decide that the rows are to be truncated as soon as no term is found within a sufficiently large search limit. (For all of the displayed rows, there is no additional term up to many orders of magnitude beyond the last term.) That way the rows are well-defined, but we are no longer guaranteed to get a partition of the integers.
The author finds the idea of partitioning the integers in this elementary yet highly nontrivial way appealing, as is the fact that the initial rows are just roughly one line long. Will this property continue to hold for large n, or if not, how will the row lengths evolve?

			The table reads:
n \ k's
0 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 19, 23, 24, 26, 27, 28, 31, 34, 68 (cf. A030700)
1 : 10, 15, 16, 17, 18, 20, 25, 29, 43, 47, 50, 52, 63, 72, 73
2 : 22, 30, 32, 33, 36, 38, 39, 40, 41, 42, 44, 45, 46, 48, 51, 53, 56, 58, 60, 61, 62, 64, 69, 71, 83, 93, 96, 108, 111, 123, 136
3 : 21, 37, 49, 67, 75, 81, 82, 87, 90, 105, 112, 121, 129
4 : 35, 59, 65, 66, 70, 74, 77, 79, 88, 98, 106, 116, 117, 128, 130, 131, 197, 205
5 : 57, 76, 78, 80, 86, 89, 91, 92, 101, 102, 104, 109, 115, 118, 122, 127, 134, 135, 164, 166, 203, 212, 237
...
The first column is A063555: least k such that 3^k has n digits '0' in base 10.
Row lengths are 23, 15, 31, 13, 18, 23, 23, 25, 16, 17, 28, ... (A305943).
Last term of the rows (i.e., largest k such that 3^k has exactly n digits 0) are (68, 73, 136, 129, 205, 237, 317, 268, 251, 276, 343, ...), A306113.
Inverse permutation is (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 23, 10, 11, 12, 13, 24, 25, 26, 27, 14, 28, 69, 38, 15, 16, 29, 17, 18, 19, 30, 39, 20, ...), not in OEIS.

Cf. A030700, A063555.
Cf. A305932 (analog for 2^k), A305924 (analog for 4^k), ..., A305929 (analog for 9^k).
Cf. A305934: powers of 3 with exactly one '0', A305943: powers of 3 with at least one '0'.

apply( A305933_row(n,M=50*n+70)=select(k->#select(d->!d,digits(3^k))==n,[0..M]), [0..10])
print(apply(t->#t,%)"\n"apply(vecmax,%)"\n"apply(t->t-1,Vec(vecsort(concat(%),,1)[1..99]))) \\ to show row lengths, last elements, and inverse permutation.

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

A131613 Numbers k such that the decimal expansion of 3^k contains no 9.

Original entry on oeis.org

0, 1, 3, 4, 5, 7, 8, 11, 12, 16, 19, 20, 21, 29, 32, 56

Offset: 1

Shyam Sunder Gupta, Sep 01 2007

I conjecture that 56 is the last term.

Numbers k such that the decimal expansion of 3^k contains no m: A030700 (m=0), A131627 (m=1), A131625 (m=2), A131629 (m=3), A131618 (m=4), A131617 (m=5), A131616 (m=6), A131615 (m=7), A131614 (m=8), this sequence (m=9).

Cf. A007377.

[n: n in [0..1000] | not 9 in Intseq(3^n) ]; // Vincenzo Librandi, May 06 2015

Join[{0}, Select[ Range@10000, FreeQ[ IntegerDigits[3^# ], 9] &]]

Adapted Mma and initial 0 added by Vincenzo Librandi, May 06 2015

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

A305934 Powers of 3 that have exactly one digit '0' in base 10.

Original entry on oeis.org

59049, 14348907, 43046721, 129140163, 387420489, 3486784401, 847288609443, 68630377364883, 328256967394537077627, 26588814358957503287787, 717897987691852588770249, 6461081889226673298932241, 1144561273430837494885949696427, 22528399544939174411840147874772641, 67585198634817523235520443624317923

Offset: 1

M. F. Hasler (following a suggestion by Zak Seidov), Jun 14 2018

Motivated by A030700: decimal expansion of 3^n contains no zeros (probably finite).
It appears that this sequence is finite. Is a(15) = 3^73 the last term?
There are no more terms through at least 3^(10^7) (which is a 4771213-digit number). It seems nearly certain that no power of 3 containing this many or more decimal digits could have fewer than two '0' digits. (Among numbers of the form 3^k with 73 < k <= 10^7, the only one having fewer than two '0' digits among its final 200 digits is 3^5028978.) - Jon E. Schoenfield, Jun 24 2018
The first 6 terms coincide with A305931: powers of 3 having at least one digit 0, with complement A238939 (within A000244: powers of 3) conjectured to be finite, too. Then, a(7..8) = A305931(9..10), etc.

Cf. A030700: decimal expansion of 3^n contains no zeros (probably finite), A238939: powers of 3 with no digit '0' in their decimal expansion, A000244: powers of 3.
Subsequence of A305931: powers of 3 having at least one '0'.
Cf. A305933: row n = { k | 3^k has n digits '0' }.

Select[3^Range[120], DigitCount[#, 10, 0] == 1 &] (* Michael De Vlieger, Jul 01 2018 *)

for(n=1,99, #select(t->!t,digits(3^n))==1&& print1(3^n","))

a(n) = 3^A305933(1,n).

A050724 Numbers k such that the decimal expansion of 3^k contains no pair of consecutive equal digits (probably finite).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 21, 22, 24, 26, 42, 66, 67, 133

Offset: 1

Patrick De Geest, Sep 15 1999

No additional terms up to 50000. - Harvey P. Dale, Nov 15 2011

No additional terms up to 100000. - Michel Marcus, Oct 16 2019

			3^133 = 2865014852390475710679572105323242035759805416923029389510561523.

Cf. A000244, A030700, A046261, A046269.

q:= n-> (s-> andmap(i-> s[i]<>s[i+1], [$1..length(s)-1]))(""||(3^n)):
select(q, [$0..200])[];  # Alois P. Heinz, Mar 07 2024

Select[Range[0,500],!MemberQ[Differences[IntegerDigits[3^#]],0]&] (* Harvey P. Dale, Nov 15 2011 *)

isok(n) = {my(d = digits(3^n), c = d[1]); for (i=2, #d, if (d[i] == c, return (0)); c = d[i];); return (1);} \\ Michel Marcus, Oct 16 2019

A305931 Powers of 3 having at least one digit '0' in their decimal representation.

Original entry on oeis.org

59049, 14348907, 43046721, 129140163, 387420489, 3486784401, 10460353203, 31381059609, 847288609443, 68630377364883, 205891132094649, 1853020188851841, 5559060566555523, 50031545098999707, 150094635296999121, 450283905890997363, 1350851717672992089, 4052555153018976267, 12157665459056928801

Offset: 1

M. F. Hasler, Jun 15 2018

The analog of A298607 for 3^k instead of 2^k.

The complement A238939 is conjectured to have only 23 elements, the largest being 3^68. Thus, all larger powers of 3 are (conjectured to be) in this sequence. Each of the subsequences "powers of 3 with exactly n digits 0" is conjectured to be finite. Provided there is at least one such element for each n >= 0, this leads to a partition of the integers, given in A305933.

Cf. A030700 = row 0 of A305933: decimal expansion of 3^n contains no zeros.
Complement (within A000244: powers of 3) of A238939: powers of 3 with no digit '0' in their decimal expansion.
Analog of A298607: powers of 2 with the digit '0' in their decimal expansion.
The first six terms coincide with the finite sequence A305934: powers of 3 having exactly one digit 0.

Select[3^Range[0,40],DigitCount[#,10,0]>0&] (* Harvey P. Dale, May 30 2020 *)

for(k=0,69, vecmin(digits(3^k))|| print1(3^k","))

select( t->!vecmin(digits(t)), apply( k->3^k, [0..40]))

A305943 Number of powers of 3 having exactly n digits '0' (in base 10), conjectured.

Original entry on oeis.org

23, 15, 31, 13, 18, 23, 23, 25, 16, 17, 28, 25, 22, 20, 18, 21, 19, 19, 18, 24, 33, 17, 17, 18, 17, 14, 21, 26, 25, 23, 24, 29, 17, 22, 18, 21, 27, 26, 20, 21, 13, 27, 24, 12, 18, 24, 16, 17, 15, 30, 24, 32, 24, 12, 16, 16, 23, 23, 20, 23, 19, 23, 10, 21, 20, 21, 23, 20, 19, 23, 23, 22, 16, 18, 20, 20, 13, 15, 25, 24, 28, 24, 21, 16, 14, 23, 21, 19, 23, 19, 27, 26, 22, 18, 27, 16, 31, 21, 18, 25, 24

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) = 23 is the number of terms in A030700 and in A238939, which include the power 3^0 = 1.

These are the row lengths of A305933. 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 vanishingly small, 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. A030700 = row 0 of A305933: k s.th. 3^k has no '0'; A238939: these powers 3^k.
Cf. A305931, A305934: powers of 3 with at least / exactly one '0'.
Cf. A020665: largest k such that n^k has no '0's.
Cf. A063555 = column 1 of A305933: least k such that 3^k has n digits '0' in base 10.
Cf. A305942 (analog for 2^k), ..., A305947, A305938, A305939 (analog for 9^k).

A305943(n,M=99*n+199)=sum(k=0,M,#select(d->!d,digits(3^k))==n)

A305943_vec(nMax,M=99*nMax+199,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(3^k)),nMax)]++);a[^-1]}

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

M. F. Hasler, Sep 26 2011

			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.

M. F. Hasler, Table of n, a(n) for n = 1..158
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, A195908, A195948, A007377, A008839, A030700, A030701, A030702, A030703, A030704, A030705, A030706.

a(n)={ my(p=sqrtint(10^n\9)-1); until( is_A052382(p^2), p=nextprime(p+2));p}

cp's OEIS Frontend

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

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A305933 Irregular table read by rows: row n >= 0 lists all k >= 0 such that the decimal representation of 3^k has n digits '0' (conjectured).

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

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A131613 Numbers k such that the decimal expansion of 3^k contains no 9.

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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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A305934 Powers of 3 that have exactly one digit '0' in base 10.

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A050724 Numbers k such that the decimal expansion of 3^k contains no pair of consecutive equal digits (probably finite).

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A305931 Powers of 3 having at least one digit '0' in their decimal representation.

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