A051833 -id:A051833 - 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

A052040 Numbers whose square is zeroless.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 54, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 72, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 96

Offset: 1

Patrick De Geest, Dec 15 1999

This sequence is infinite, since 33...334^2 = 11...11155...556, for example. This answers an open problem stated in HAKMEM. - Karl W. Heuer, Aug 19 2015

			From _Jon E. Schoenfield_, Aug 16 2021: (Start)
31 is a term: 31^2 = 961 has no 0's among its digits.
32 is not a term, because 32^2 = 1024. (End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Michael Beeler, R. William Gosper, and Rich Schroeppel, "HAKMEM" Item 36, Memo 239, Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, Mass., 1972.

Squares: A052041, A052042, A052043. Cubes: A051750, A051751, A051832, A051833.

Cf. A052382.

[n: n in [1..100] | not 0 in Intseq(n^2)]; // Vincenzo Librandi, Feb 22 2015

Select[Range[0, 100], DigitCount[#^2, 10, 0]==0 &] (* Vincenzo Librandi, Feb 22 2015 *)

is(n)=vecmin(digits(n^2))>0 \\ Charles R Greathouse IV, Oct 11 2013

A052043 Squares of primes lacking the digit zero in their decimal expansion.

Original entry on oeis.org

4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 3481, 3721, 4489, 5329, 6241, 6889, 7921, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 24649, 26569, 27889, 29929, 32761, 36481, 37249, 44521, 49729, 51529, 52441, 54289

Offset: 1

Patrick De Geest, Dec 15 1999

Intersection of A052382 and A001248; A168046(a(n))*A064911(a(n))*A010052(a(n)) = 1. - Reinhard Zumkeller, Dec 01 2009

Zak Seidov, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Zerofree - _Reinhard Zumkeller_, Dec 01 2009

Squares: A052042, A052040, A052041.

Cubes: A051750, A051751, A051832, A051833.

Select[Prime[Range[100]]^2,DigitCount[#,10,0]==0&] (* Harvey P. Dale, Mar 18 2012 *)

is(n)=my(d=digits(n));vecsort(d)[1]&&issquare(n,&n)&&isprime(n) \\ Charles R Greathouse IV, Jun 05 2013

a(n) = A052042(n)^2. - R. J. Mathar, Jul 23 2025

A052042 Primes that lack the digit zero in the decimal expansion of their squares.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 59, 61, 67, 73, 79, 83, 89, 107, 109, 113, 127, 131, 137, 139, 157, 163, 167, 173, 181, 191, 193, 211, 223, 227, 229, 233, 239, 263, 269, 271, 277, 281, 293, 307, 311, 313, 337, 359, 367, 373, 379, 383, 389, 409, 419, 421, 431

Offset: 1

Patrick De Geest, Dec 15 1999

			The primes 47, 53 and 71 are not in the sequence because 47^2=2209, 53^2=2809 and 71^2=5041 contain zeros in their decimal representation.

Zak Seidov, Table of n, a(n) for n = 1..10000

Squares: A052043, A052040, A052041. Cubes: A051750, A051751, A051832, A051833.

Cf. A245576.

fQ[n_] := DigitCount[n^2][[-1]] == 0; Select[Prime@ Range@ 80, fQ] (* Robert G. Wilson v, Aug 22 2012 *)

{p=2;for(k=1,10^2,if(vecmin(digits(p^2))>0,
print1(p", "));p=nextprime(1+p))}\\ Zak Seidov, Dec 24 2014

a(n) = sqrt(A052043(n)). - Zak Seidov, Dec 27 2014

A051750 Primes whose cubes lack zeros.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 53, 61, 71, 83, 97, 113, 137, 139, 151, 157, 167, 173, 179, 181, 191, 197, 211, 233, 239, 241, 251, 257, 263, 277, 283, 293, 307, 331, 337, 347, 353, 359, 373, 379, 383, 389, 409, 421, 433, 457, 461, 463, 499, 503

Offset: 1

G. L. Honaker, Jr., Dec 07 1999

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

Cf. A000040, A000578.

Cubes: A052044, A052045, A051751, A051832, A051833. Squares: A052040, A052041, A052042, A052043.

Select[Prime[Range[100]],FreeQ[IntegerDigits[#^3],0]&] (* Harvey P. Dale, Jan 06 2017 *)

isok(n) = isprime(n) && vecmin(digits(n^3)); \\ Michel Marcus, Jan 06 2014

More terms from Michel Marcus, Jan 06 2014

A051832 Numbers k such that (210^(5k) - 10^(4k) + 210^(3k) + 10^(2k) + 10^k + 1)/3 is prime.

Original entry on oeis.org

0, 1, 7, 133

Offset: 1

G. L. Honaker, Jr., Dec 11 1999

The Baxter-Hickerson function provides a number whose cube lacks zeros.
The next term is > 4400. - Jason Earls, Sep 10 2005
The next term is > 20000 (found using pfgw64). - Patrick De Geest, Jul 22 2012

Ed Pegg, Jr., Fun with numbers.
Eric Weisstein's World of Mathematics, Baxter-Hickerson Function.

Cubes: A052044, A052045, A051750, A051751, A051833. Squares: A052040, A052041, A052042, A052043.

f := n->(2*10^(5*n) - 10^(4*n) + 2*10^(3*n) + 10^(2*n) + 10^n + 1)/3;

is(n)=isprime((2*10^(5*n)-10^(4*n)+2*10^(3*n)+10^(2*n)+10^n+1)/3) \\ Charles R Greathouse IV, Feb 17 2017

A052044 Numbers k such that k^3 lacks the digit zero in its decimal expansion.

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, 32, 33, 35, 36, 38, 39, 41, 44, 45, 46, 49, 51, 53, 54, 55, 56, 57, 58, 61, 62, 64, 65, 66, 68, 71, 72, 75, 76, 77, 78, 81, 82, 83, 85, 88, 91, 92, 95, 96, 97, 98, 104, 105, 108, 111

Offset: 1

Patrick De Geest, Dec 15 1999

This sequence is infinite since A052427 is a subsequence. - Amiram Eldar, Nov 23 2020

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Logarithmic scatterplot of (n, a(n+1)-a(n)) for n = 1..1000000

Cubes: A052045, A051750, A051751, A051832, A051833, A052427.

Squares: A052040, A052041, A052042, A052043.

Select[Range[120],DigitCount[#^3,10,0]==0&] (* Harvey P. Dale, Oct 24 2011 *)

is(n)=vecmin(digits(n^3))>0 \\ Charles R Greathouse IV, Oct 11 2013

a(n) = A052045(n)^(1/3). - Amiram Eldar, Nov 23 2020

A051751 Cubes arising in A051750.

Original entry on oeis.org

8, 27, 125, 343, 1331, 2197, 4913, 6859, 12167, 24389, 29791, 68921, 148877, 226981, 357911, 571787, 912673, 1442897, 2571353, 2685619, 3442951, 3869893, 4657463, 5177717, 5735339, 5929741, 6967871, 7645373, 9393931, 12649337

Offset: 1

G. L. Honaker, Jr., Dec 07 1999

Cubes: A052044, A052045, A051750, A051832, A051833. Squares: A052040, A052041, A052042, A052043.

A030078 INTERSECT A052382. - R. J. Mathar, Mar 23 2007

A052045 Cubes lacking the digit zero in their decimal expansion.

Original entry on oeis.org

1, 8, 27, 64, 125, 216, 343, 512, 729, 1331, 1728, 2197, 2744, 3375, 4913, 5832, 6859, 9261, 12167, 13824, 15625, 17576, 19683, 21952, 24389, 29791, 32768, 35937, 42875, 46656, 54872, 59319, 68921, 85184, 91125, 97336, 117649, 132651, 148877

Offset: 1

Patrick De Geest, Dec 15 1999

This sequence is infinite since A052427(n)^3 is a term for all n>=0. - Amiram Eldar, Nov 23 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
Eric Weisstein's World of Mathematics, Zerofree [From _Reinhard Zumkeller_, Dec 01 2009]

Cubes: A052044, A051750, A051751, A051832, A051833, A052427.

Squares: A052040, A052041, A052042, A052043.

select(t -> not has(convert(t,base,10),0), [seq(m^3,m=1..10^3)]); # Robert Israel, Aug 24 2014

Select[Range[53]^3, DigitCount[#, 10, 0] == 0 &] (* Amiram Eldar, Nov 23 2020 *)

lista(nn) = {for (n=1, nn, if (vecmin(digits(cub=n^3)), print1(cub, ", ")););} \\ Michel Marcus, Aug 25 2014

A052045 = [n**3 for n in range(1,10**5) if not str(n**3).count('0')]
# Chai Wah Wu, Aug 24 2014

Intersection of A052382 and A000578; A168046(a(n))*A010057(a(n)) = 1. - Reinhard Zumkeller, Dec 01 2009

a(n) = A052044(n)^3. - Amiram Eldar, Nov 23 2020

A052427 Baxter-Hickerson numbers.

Original entry on oeis.org

2, 64037, 6634003367, 666334000333667, 66663334000033336667, 6666633334000003333366667, 666666333334000000333333666667, 66666663333334000000033333336666667

Offset: 0

Eric W. Weisstein

nonn

From Amiram Eldar, Nov 23 2020: (Start)
Named after Lew Baxter and Dean Hickerson.
Pegg (1999) conjectured that the sequence of zeroless cubes (A052045) is finite. On April 19, 1999, Hickerson gave the counterexample: if n == 2 (mod 3) and n >= 5, then the cube of (2*10^(5*n) - 10^(4*n) + 17*10^(3*n-1) + 10^(2*n) + 10^n - 2)/3 is zeroless. Three days later, Baxter gave a simpler variation which is valid for all n>=0 and is given in the Formula section. (End)

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005. See p. 109.

Amiram Eldar, Table of n, a(n) for n = 0..200
Lew Baxter, Cubes lacking zeros, sci.math newsgroup, April 22, 1999.
Ed Pegg, Jr., Cube conjecture, sci.math newsgroup, April 18, 1999.
Ed Pegg, Jr., Fun with Numbers, mathpuzzle websize.
Eric Weisstein's World of Mathematics, Baxter-Hickerson Function.

Subsequence of A052044.

Cf. A016789, A051832, A051833, A052045.

a(0) = 2, and 2^3 = 8 is zeroless.
a(1) = 64037, and 64037^3 = 262598918898653 is zeroless.

a[n_] := (2*10^(5*n) - 10^(4*n) + 2*10^(3*n) + 10^(2*n) + 10^n + 1)/3; Array[a, 10, 0] (* Amiram Eldar, Nov 23 2020 *)

a(n) = (2*10^(5*n) - 10^(4*n) + 2*10^(3*n) + 10^(2*n) + 10^n + 1)/3 (Baxter, 1999). - Amiram Eldar, Nov 23 2020

Offset changed to 0 by Amiram Eldar, Nov 23 2020

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A052041 Squares lacking the digit zero in their decimal expansion.

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A052040 Numbers whose square is zeroless.

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A052043 Squares of primes lacking the digit zero in their decimal expansion.

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A052042 Primes that lack the digit zero in the decimal expansion of their squares.

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A051750 Primes whose cubes lack zeros.

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A051832 Numbers k such that (2*10^(5*k) - 10^(4*k) + 2*10^(3*k) + 10^(2*k) + 10^k + 1)/3 is prime.

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A052044 Numbers k such that k^3 lacks the digit zero in its decimal expansion.

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A051751 Cubes arising in A051750.

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A051832 Numbers k such that (210^(5k) - 10^(4k) + 210^(3k) + 10^(2k) + 10^k + 1)/3 is prime.