A252484 -id:A252484 - cp's OEIS frontend

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

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)

A156954 Integers N such that by inserting + or - or * or / or ^ between each of their digits, without any grouping parentheses, you can get N (the ambiguous a^b^c is avoided).

Original entry on oeis.org

736, 2592, 11664, 15617, 15618, 15622, 15624, 15632, 15642, 15645, 15656, 15662, 15667, 15698, 17536, 27639, 32785, 39363, 39369, 45947, 46633, 46644, 46648, 46655, 46660, 46663, 117635, 117638, 117639, 117642, 117643, 117647, 117650

Offset: 1

Jean-Marc Falcoz, Feb 19 2009

The single-digit numbers 0, ..., 9 are here excluded by convention although they also ("voidly") satisfy the definition and therefore logically should be terms of this sequence. This is in contrast to the Friedman numbers A036057 where the construction also allows concatenation of digits but then of course has to exclude the case where only concatenation of the digits is used, which excludes the single-digit terms. - M. F. Hasler, Jan 07 2015

A subset of the orderly Friedman numbers A080035. - M. F. Hasler, Jan 04 2015

			736 = 7 + 3^6.
2592 = 2^5*9^2.
11664 = 1*1*6^6/4.
15617 = 1*5^6 - 1 - 7.
For more examples, see the link to "decompositions".

Giovanni Resta, Table of n, a(n) for n = 1..423 (terms < 10^8)
Giovanni Resta, Decompositions for terms < 10^8

Cf. A036057, A080035, A252484.

is(n,o=Vecsmall("*+-^/"))={v=Vecsmall(Str(n,n\10)); forstep(i=#v,3,-2,v[i]=v[i\2+1]); n>9 && forvec(s=vector(#v\2,i,[1,#o-(v[i*2+1]==48)]), for(i=1,#s,94==(v[2*i]=o[s[i]])&&i>1&&s[i-1]==4&&next(2));n==eval(Strchr(v))&&return(1))}

Edited by M. F. Hasler, Jan 04 2015

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

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

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A156954 Integers N such that by inserting + or - or * or / or ^ between each of their digits, without any grouping parentheses, you can get N (the ambiguous a^b^c is avoided).

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

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Mathematica

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

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PARI

PARI

Python

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