A253646 -id:A253646 - cp's OEIS frontend

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

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)

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

Original entry on oeis.org

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)

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)

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

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

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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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Maple

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

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Maple

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

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