A253644 -id:A253644 - cp's OEIS frontend

A253645 Primes p such that p^k is zeroless for k=0,...,5.

2, 3, 5, 13, 17, 23, 31, 137, 233, 337, 383, 719, 971, 1291, 1663, 1777, 3623, 6113, 12589, 15733, 15791, 17729, 22637, 44623, 48989, 61379, 65777, 72379, 82129, 91331, 99559, 132859, 133733, 163633, 226129, 239539, 346391, 352133, 394223, 415379, 428531, 485113, 518233, 546523

Offset: 1

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

These are the primes in A253644. This is a subsequence of A253110 (k<=4) and contains A253646 (k <= 6) as a subsequence.

Motivated by A253646, i.e., the observation that many small primes satisfy this condition for k <= 5 (52 terms below 10^6) but only very few satisfy it for k <= 6 (only 2 terms between 20 and 2*10^9).

Giovanni Resta, Table of n, a(n) for n = 1..5300 (first 1200 terms from Zak Seidov)

Cf. A253110, A253644, A253646.

Select[Prime[Range[46000]],Count[Flatten[IntegerDigits/@(#^Range[5])],0] == 0&] (* Harvey P. Dale, Jan 13 2015 *)

forprime(p=0,,forstep(k=5,1,-1,vecmin(digits(p^k))||next(2));print1(p","))

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)

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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A253645 Primes p such that p^k is zeroless for k=0,...,5.

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

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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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