A124648 -id:A124648 - cp's OEIS frontend

A124649 Numbers n such that n^i (i=1..8) are all zeroless.

1, 2, 3, 6, 68

Offset: 1

Zak Seidov, Dec 22 2006

No other terms < 10^6. Is the sequence finite?
Subsequence of A124648. No other terms below 10^8. - Michel Marcus, Oct 11 2013
a(6) > 3.3*10^16, if it exists. - Giovanni Resta, Sep 07 2018

			68^i (i=1..8)= 68, 4624, 314432, 21381376, 1453933568, 98867482624, 6722988818432, 457163239653376 all zeroless.

Cf. A104264.

zLessQ[n_]:=AllTrue[n^Range[8],FreeQ[IntegerDigits[#],0]&]; Select[Range[70],zLessQ] (* Harvey P. Dale, Sep 27 2023 *)

isok(n) = {for (i = 1, 8, if (! vecmin(digits(n^i)), return (0));); return (1);} \\ Michel Marcus, Oct 11 2013

A253647 Numbers n such that n^k is zeroless for k=0,...,6.

Original entry on oeis.org

1, 2, 3, 5, 6, 14, 17, 68, 76, 96, 188, 483, 518, 582, 736, 786, 1331, 1414, 3944, 4214, 6112, 6676, 8256, 8583, 8686, 9738, 15314, 15483, 33736, 44712, 48989, 61562, 71689, 78512, 93711, 121568, 187791, 239477, 292958, 315426, 545866, 763142, 792612, 1391739

Offset: 1

M. F. Hasler, Jan 07 2015

Contains A124648 as a subsequence. Primes in this sequence are listed in A253646.

There are 55 terms below 10^7. Conjectured to be finite.

Giovanni Resta, Table of n, a(n) for n = 1..191 (terms < 3.3*10^16)

Cf. A052382, A104264, A124648, A124649, A253645, A253646.

Select[Range[14*10^5],Count[Flatten[IntegerDigits/@(#^Range[ 0,6])],0] == 0&] (* Harvey P. Dale, Apr 29 2018 *)

is_A253647(n,K=6)=!forstep(k=K,1,-1,vecmin(digits(n^k))||return)

A252484 Numbers m such that m^k is zeroless for k=1,...,4.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 11, 13, 14, 17, 21, 23, 24, 26, 27, 28, 31, 36, 39, 41, 46, 56, 58, 61, 62, 66, 68, 72, 76, 82, 83, 88, 91, 92, 96, 121, 122, 129, 137, 146, 154, 161, 162, 166, 167, 168, 183, 186, 188, 189, 211, 231, 233, 244, 256, 262, 264, 268, 277, 278, 289, 296, 337, 373, 374, 376, 382, 383

Offset: 1

M. F. Hasler, Jan 07 2015

See A253110 for the primes in this sequence. See A253644 for the subsequence including k=5.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A052382, A253643 (k <= 3), A253644 (k <= 5), A253645 (primes, k <= 5), A253647 (k <= 6), A253646 (primes, k <= 6), A124648 (k <= 7), A124649 (k <= 8).

Cf. A104264.

filter:= proc(n)
local j;
for j from 0 to 4 do
  if has(convert(n^j,base,10),0) then return false fi
od:
true
end proc:
select(filter, [$1..1000]); # Robert Israel, Jan 15 2015

Select[Range[400],Union[DigitCount[#^Range[4],10,0]]=={0}&] (* Harvey P. Dale, Aug 01 2020 *)

is_A252484(n,K=4)=!forstep(k=K,1,-1,vecmin(digits(n^k))||return)

A253644 Numbers n such that n^k is zeroless for k=0,...,5.

Original entry on oeis.org

1, 2, 3, 5, 6, 13, 14, 17, 23, 24, 26, 31, 58, 62, 66, 68, 72, 76, 88, 96, 137, 168, 188, 233, 244, 262, 264, 296, 337, 376, 382, 383, 483, 488, 511, 514, 518, 519, 582, 628, 719, 736, 786, 816, 822, 928, 938, 971, 978, 1122, 1178, 1291, 1331, 1392, 1413, 1414, 1663, 1777

Offset: 1

M. F. Hasler, Jan 07 2015

A subsequence of A252484 (analog for k <= 4) which contains A253647 (analog including k = 6) as a subsequence. Primes in this sequence are listed in A253645.

Conjectured to be finite.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A052382, A104264, A124648, A124649, A253645, A253646, A253647.

filter:= proc(x)
  local k;
  for k from 1 to 5 do
    if has(convert(x^k,base,10),0) then return false fi
  od:
  true
end proc:
select(filter, [$1..10000]); # Robert Israel, Jan 07 2015

Select[Range[2000],Count[Flatten[IntegerDigits/@(#^Range[5])],0]==0&] (* Harvey P. Dale, Jun 10 2017 *)

is_A253644(n,K=5)=!forstep(k=K,1,-1,vecmin(digits(n^k))||return)

A253646 Primes p such that p^k is zeroless for k=1,...,6.

Original entry on oeis.org

2, 3, 5, 17, 48989, 5453971, 61636943111479, 128359315177123, 884785266899689, 1116777231836989

Offset: 1

Zak Seidov and M. F. Hasler, Jan 07 2015

Primes in A253647; both sequences are conjectured to be finite.
The motivation for this sequence lies in the fact that many small primes satisfy the restriction up to k=5 (there are 52 terms below 10^6, cf. A253645), but including k=6 makes the sequence much sparser, with only one term between 17 and 5*10^6, and only one more term below 2*10^9.
The terms 2, 3 and 5 seem to be the only primes in A124648, i.e., satisfy the restriction up to k=7.
a(7) > 10^11. - Chai Wah Wu, Jan 10 2015
a(11) > 3.3*10^16. - Giovanni Resta, Sep 06 2018

Cf. A253647, A253645, A124648.

Select[Prime[Range[10^7]],Count[Flatten[IntegerDigits/@(#^Range[6])],0] == 0&] (* Harvey P. Dale, May 26 2016 *)

forprime(p=0,,forstep(k=6,1,-1,vecmin(digits(p^k))||next(2));print1(p","))

from sympy import isprime
A253646_list = [2]
for i in range(1,10**6,2):
    if not '0' in str(i):
        m = i
        for k in range(5):
            m *= i
            if '0' in str(m):
                break
        else:
            if isprime(i):
                A253646_list.append(i) # Chai Wah Wu, Jan 10 2015

a(7)-a(10) from Giovanni Resta, Sep 03 2018

A253643 Numbers n such that n^k is zeroless for k=0,...,3.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 17, 18, 19, 21, 23, 24, 25, 26, 27, 28, 29, 31, 35, 36, 38, 39, 41, 44, 46, 54, 56, 57, 58, 61, 62, 65, 66, 68, 72, 75, 76, 77, 81, 82, 83, 85, 88, 91, 92, 96, 111, 113, 114, 119, 121, 122, 125, 129, 132, 133, 136, 137, 139, 146, 154, 156, 157, 158, 161

Offset: 1

M. F. Hasler, Mar 09 2015

See A252484 for the subsequence of numbers having this property up to k=4.

Cf. A052382, A252484 (k <= 4), A253644 (k <= 5), A253645 (primes, k <= 5), A253647 (k <= 6), A253646 (primes, k <= 6), A124648 (k <= 7), A124649 (k <= 8).

Cf. A104264.

Select[Range[200],AllTrue[#^Range[3],DigitCount[#,10,0]==0&]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 15 2015 *)

is_A253643(n,K=3)=!forstep(k=K,1,-1,vecmin(digits(n^k))||return)

for n in range(100):
  s1,s2,s3 = str(n),str(n**2),str(n**3)
  if s1.find('0') + s2.find('0') + s3.find('0') == -3:
    print(n,end=', ') # Derek Orr, Mar 09 2015

A358340 a(n) is the smallest n-digit number whose fourth power is zeroless.

Original entry on oeis.org

1, 11, 104, 1027, 10267, 102674, 1026708, 10266908, 102669076, 1026690113, 10266901031, 102669009704, 1026690096087, 10266900960914, 102669009608176, 1026690096080369, 10266900960803447, 102669009608034434, 1026690096080341627, 10266900960803409734, 102669009608034097731, 1026690096080340972491

Offset: 1

Mohammed Yaseen, Nov 10 2022

It has been proved that there exist infinitely many zeroless squares and cubes but there is apparently no proof for 4th powers, 5th powers, etc.

This sequence approaches the decimal expansion of 9000^(-1/4). Similar sequences of other small powers k seem to approach the decimal expansion of (9*10^(k-1))^(-1/k).

Michael S. Branicky, Table of n, a(n) for n = 1..69
Eric Weisstein's World of Mathematics, Zerofree

Cf. A052382, A052040, A052044.

Cf. A253643, A252484, A253644, A253647, A124648, A124649.

a(n) = my(x=10^(n-1)); while(! vecmin(digits(x^4)), x++); x; \\ Michel Marcus, Nov 10 2022

a(n) = { my(s = sqrtnint(10^(4*n - 3) \ 9, 4)); for(i = s, oo, c = i^4; if(vecmin(digits(c)) > 0, return(i) ) ) } \\ David A. Corneth, Nov 10 2022

from itertools import count
from sympy import integer_nthroot
def a(n):
    start = integer_nthroot(int("1"*(4*(n-1)+1)), 4)[0]
    return next(i for i in count(start) if "0" not in str(i**4))
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Nov 10 2022

a(n) ~ 10^(n + 1/4) / sqrt(3).

More terms from David A. Corneth, Nov 10 2022

A385332 Integers k such that the set {k, k^2, ..., k^9} contains at least 8 zeroless numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 11, 17, 68, 121, 786

Offset: 1

Gonzalo Martínez, Jun 25 2025

1, 2, 3, 6 and 68 are the only integers less than 3.3*10^16 such that {k, k^2,..., k^8} are all zeroless (See A124649), but of them 1^9, 2^9 and 3^9 are the only zeroless ones. If we weaken the condition and ask that 8 of the 9 numbers in the set {k, k^2,..., k^9} are zeroless, then more numbers appear that satisfy this property.

Is a(10) = 786 the largest term?

			5 is a term, since the first nine powers of 5 are: 5, 25, 125, 625, 3125, 15625, 78125, 390625 and 1953125, where 8 of the 9 are zeroless.

Cf. A124648, A124649.

zless:= n -> not has(convert(n,base,10),0):
filter:= proc(n) local i; nops(select(zless, [seq(n^i,i=1..9)]))>=8 end proc:
select(filter, [$1..1000]); # Robert Israel, Jun 26 2025

Select[Range[10^5],Total[Boole/@Positive/@Min/@IntegerDigits/@(#^Range[9])]>7&] (* James C. McMahon, Jul 01 2025 *)

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A124649 Numbers n such that n^i (i=1..8) are all zeroless.

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A253647 Numbers n such that n^k is zeroless for k=0,...,6.

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A252484 Numbers m such that m^k is zeroless for k=1,...,4.

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A253644 Numbers n such that n^k is zeroless for k=0,...,5.

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A253646 Primes p such that p^k is zeroless for k=1,...,6.

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A253643 Numbers n such that n^k is zeroless for k=0,...,3.

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A358340 a(n) is the smallest n-digit number whose fourth power is zeroless.

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A385332 Integers k such that the set {k, k^2, ..., k^9} contains at least 8 zeroless numbers.

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