A104264 -id:A104264 - cp's OEIS frontend

A052041 Squares lacking the digit zero in their decimal expansion.

1, 4, 9, 16, 25, 36, 49, 64, 81, 121, 144, 169, 196, 225, 256, 289, 324, 361, 441, 484, 529, 576, 625, 676, 729, 784, 841, 961, 1156, 1225, 1296, 1369, 1444, 1521, 1681, 1764, 1849, 1936, 2116, 2916, 3136, 3249, 3364, 3481, 3721, 3844, 3969, 4225, 4356

Offset: 1

Patrick De Geest, Dec 15 1999

This sequence is infinite: see A075415 or A102807 for a constructive proof.

Intersection of A052382 and A000290; A168046(a(n))*A010052(a(n))=1. - Reinhard Zumkeller, Dec 01 2009

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Zerofree [From _Reinhard Zumkeller_, Dec 01 2009]

Cf. A104264, A102265, A102266.

Squares: A052040, A052042, A052043. Cubes: A051750, A051751, A051832, A051833.

Select[Range[66]^2, FreeQ[IntegerDigits[#],0]==True &] (* Jayanta Basu, May 25 2013 *)

a(n) = A052040(n)^2. - R. J. Mathar, Jul 23 2025

A124648 Numbers n such that n^i (i=1..7) are all zeroless.

Original entry on oeis.org

1, 2, 3, 5, 6, 68, 76, 3944, 15483

Offset: 1

Zak Seidov, Dec 22 2006

No other terms < 10^8. - Michel Marcus, Oct 11 2013
No other terms < 10^13. - Charles R Greathouse IV, Oct 14 2013
Subsequence of A253647, the analog with i <= 6 instead of 7. Conjectured to be finite. - M. F. Hasler, Jan 07 2015
a(10) > 3.3*10^16, if it exists. - Giovanni Resta, Sep 06 2018

			15483^i (i=1..7) = 15483, 239723289, 3711635683587, 57467255288977521, 889765513639238957643, 13776239447676336781186569, 213297515368372722383111647827 all zeroless.

Cf. A253647, A104264, A124649.

Select[Range[10^6], FreeQ[Union[IntegerDigits[ # ],IntegerDigits[ #^2],IntegerDigits[ #^3],IntegerDigits[ #^4],IntegerDigits[ #^5],IntegerDigits[ #^6],IntegerDigits[ #^7]],0]&]
Select[Range[15500],FreeQ[Flatten[IntegerDigits/@(#^Range[7])],0]&] (* Harvey P. Dale, Jan 14 2024 *)

isok(n) = {for (i = 1, 7, if (! vecmin(digits(n^i)), return (0));); return (1);} \\ Michel Marcus, Oct 11 2013

\\ Script for checking for large (> 10^9) members:
is(n)=for(i=1,7,if(vecmin(digits(n^i))==0, return(0))); 1
bad(n,d)=for(k=1,d,if(n%10==0,return(1));n\=10);0
good7(n,d)=my(t=1);for(i=1,7,if(bad(lift(t*=n),d),return(0)));1
left(d)=my(v=List(),m=10^d);for(i=0,10^d-1, if(good7(Mod(i,m),d), listput(v,i)));Vec(v)
diff(v)=vector(#v-1,i,v[i+1]-v[i])
L=left(9);D=diff(concat(L,10^9+L[1]));forstep(n=L[1],1e12,D, if(is(n),print(n))) \\ Charles R Greathouse IV, Oct 14 2013

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

Original entry on oeis.org

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)

A104265 Smallest n-digit square with no zero digits.

Original entry on oeis.org

1, 16, 121, 1156, 11236, 111556, 1115136, 11115556, 111112681, 1111155556, 11111478921, 111111555556, 1111118377216, 11111115555556, 111111226346761, 1111111155555556, 11111112515384644, 111111111555555556, 1111111112398242916, 11111111115555555556, 111111111113333185156, 1111111111155555555556

Offset: 1

Reinhard Zumkeller, Feb 26 2005

			a(3) = Min{121, 144, 169, 196, ....} = 121.

Chai Wah Wu, Table of n, a(n) for n = 1..362

Cf. A102807, A104266, A104264, A052041.

f[n_] := Block[{k = Ceiling[ Sqrt[10^n]]}, While[ Union[ IntegerDigits[ k^2]][[1]] == 0, k++ ]; k^2]; Table[ f[n], {n, 0, 20}] (* Robert G. Wilson v, Mar 02 2005 *)
snds[n_]:=Module[{c=Ceiling[Sqrt[FromDigits[Join[PadRight[{},n-1,1], {0}]]]]^2},While[DigitCount[c,10,0]>0,c=(1+Sqrt[c])^2];c]; Array[ snds,22] (* Harvey P. Dale, Jun 12 2020 *)

from sympy import integer_nthroot
def A104265(n):
    m, a = integer_nthroot((10**n-1)//9,2)
    if not a:
        m += 1
    k = m**2
    while '0' in str(k):
        m += 1
        k += 2*m-1
    return k # Chai Wah Wu, Mar 24 2020

From Chai Wah Wu, Mar 24 2020: (Start)
a(n) >= (10^n-1)/9.
a(2n) = (10^n+2)^2/9 = A102807(n). Proof: the smallest 2n-digit number without zero digits is (10^(2n)-1)/9. ((10^n-1)/3)^2 = (10^(2n)-2*10^n+1)/9 < (10^(2n)-1)/9 for n >= 1. Thus a(2n) > ((10^n-1)/3)^2. The next square is ((10^n+2)/3)^2 = (10^(2n)-1)/9 + 4*(10^(n)-1)/9 + 1, i.e. it is n 1's followed by n-1 5's followed by the digit 6, and has no zero digits.
(End)

More terms from Robert G. Wilson v, Mar 02 2005
Two more terms from Jon E. Schoenfield, Mar 29 2015
a(21)-a(22) from Chai Wah Wu, Mar 24 2020

A104266 Largest n-digit square with no zero digits.

Original entry on oeis.org

9, 81, 961, 9216, 99856, 978121, 9998244, 99321156, 999887641, 9978811236, 99999515529, 999332111556, 9999995824729, 99978881115136, 999999961946176, 9999333211115556, 99999999356895225, 999978918111112681, 9999999986285964964, 99999333321111155556

Offset: 1

Reinhard Zumkeller, Feb 26 2005

See Formula section for exact formula for terms whose index n is divisible by 4, and upper bounds for other cases; see Links for additional information on those other cases. - Jon E. Schoenfield, Mar 30 2015

			a(3) = Max{...., 729, 784, 841, 961} = 961.

Jon E. Schoenfield, Table of n, a(n) for n = 1..100
Jon E. Schoenfield, Odd-indexed terms with central digits aligned
Jon E. Schoenfield, Patterns and upper bound for terms for which n mod 4 = 2

Cf. A104265, A104264, A052041.

f:= proc(n) local r;
  r:= floor(sqrt(10^n));
  while has(convert(r^2,base,10),0) do r:= r-1 od:
r^2
end proc:
seq(f(n),n=1..24); # Robert Israel, Mar 29 2015

f[n_] := Block[{k = Floor[ Sqrt[10^n]]}, While[ Union[ IntegerDigits[ k^2]][[1]] == 0, k-- ]; k^2]; Table[ f[n], {n, 18}] (* Robert G. Wilson v, Mar 03 2005 *)

a(n)=k=floor(sqrt(10^n));while(k,if(type(k)=="t_INT"&&vecmin(digits(k^2)), return(k^2));k--)
vector(20,n,a(n)) \\ Derek Orr, Mar 29 2015

From Jon E. Schoenfield, Mar 31 2015: (Start)
If n is divisible by 4, then a(n) = (10^(n/2) - ceiling(10^(n/4)/3))^2;
otherwise, if n is even, then a(n) < 10^(n) * (1 - (10^-((n-2)/4))* 2 / sqrt(90/1.000000000001026)) (see Links for derivation), except that a(2) = 81.
If n is odd, then a(n) ~ (floor(10^(n/2)))^2. (Although (floor(10*(n/2)))^2 gives an obvious upper bound for a(n) for all n, it seems to be a much tighter upper bound for odd values of n.) (End)

More terms from Robert G. Wilson v, Mar 03 2005

More terms from Jon E. Schoenfield, Mar 29 2015

A104317 Number of n-digit squares with no zero digits, having roots containing at least one zero.

Original entry on oeis.org

0, 0, 0, 0, 14, 35, 186, 446, 2151, 5579, 22348, 58927, 216816, 583410

Offset: 1

Reinhard Zumkeller, Mar 01 2005

a(n) < A104264(n).

			a(5) = #{11236=106, 11449=107^2, 11664=108^2, 11881=109^2,
41616=204^2, 42436=206^2, 42849=207^2, 43264=208^2, 43681=209^2,
92416=304^2, 93636=306^2, 94249=307^2, 94864=308^2, 95481=309^2} = 14.

Cf. A104316, A104315, A104264.

A124650 Number of n-digit numbers m such that m and m^10 are zeroless.

Original entry on oeis.org

3, 14, 42, 141, 399, 1274, 4123, 12688, 39604, 124987, 392908, 1233059, 3869012

Offset: 1

Zak Seidov, Dec 22 2006

			a(3) = 42 because #{113, 118, 125, 142, 144, 166, 172, 176, 186, 187, 195, 228, 244, 263, 268, 296, 315, 334, 357, 367, 376, 383, 387, 388, 396, 426, 467, 527, 563, 573, 574, 662, 764, 773, 783, 788, 863, 883, 885, 892, 976, 992} = 42
Corresponding zeroless 10th powers are:
{339456738992222314849,523383555379856794624,931322574615478515625,
... 318895612267497741289677389824,784328825964927423585898725376,
922819411957263335393616461824}.

Cf. A104264.

from itertools import product
def A124650(n):
    return sum(1 for s in product('123456789',repeat=n) if '0' not in str(int(''.join(s))**10)) # Chai Wah Wu, Mar 24 2020

a(7)-a(8) from Donovan Johnson, Feb 17 2010
a(9)-a(10) from Chai Wah Wu, Mar 24 2020
a(11)-a(13) from Giovanni Resta, Mar 27 2020

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A052041 Squares lacking the digit zero in their decimal expansion.

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A124648 Numbers n such that n^i (i=1..7) are all zeroless.

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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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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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A104265 Smallest n-digit square with no zero digits.

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A104266 Largest n-digit square with no zero digits.

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