A052044 -id:A052044 - cp's OEIS frontend

A051833 Primes of form (210^(5n) - 10^(4n) + 210^(3n) + 10^(2n) + 10^n + 1)/3.

Original entry on oeis.org

2, 64037, 66666663333334000000033333336666667

Offset: 1

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

The Baxter-Hickerson function provides a number whose cube lacks zeros.

Next term has 665 digits and is in b-file.

Amiram Eldar, Table of n, a(n) for n = 1..4
Ed Pegg, Jr., Fun with numbers.
Eric Weisstein's World of Mathematics, Baxter-Hickerson Function.

Cubes: A051750, A051751, A051832, A052044, A052045, A052427.

Squares: A052040, A052041, A052042, A052043.

Select[Table[(2*10^(5*n) - 10^(4*n) + 2*10^(3*n) + 10^(2*n) + 10^n + 1)/3, {n, 0, 150}], PrimeQ] (* Amiram Eldar, Jul 18 2025 *)

a(n) = A052427(A051832(n)). - Amiram Eldar, Jul 18 2025

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

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

A116978 Cubes whose multiplicative digital root is also a cube.

Original entry on oeis.org

0, 1, 8, 64, 125, 343, 512, 1000, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 13824, 15625, 17576, 19683, 21952, 27000, 32768, 35937, 39304, 42875, 46656, 50653, 54872, 59319, 64000, 68921, 74088, 79507, 85184, 91125, 97336, 103823, 110592, 117649

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), Apr 01 2006

Presumably a(n) ~ n^3. Up to 10^6 there are 14 missing cubes, up to 10^9 there are 36, up to 10^12 there are 81, up to 10^15 there are 155, up to 10^18 there are 267, up to 10^21 there are 517, and up to 10^24 there are 846. - Charles R Greathouse IV, Nov 17 2015

Nathaniel Johnston, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Multiplicative Digital Root.

Cf. A000578, A031347, A117678, A052044.

A007954 := proc(n) return mul(d, d=convert(n, base, 10)): end: A116978 := proc(n) option remember: local k,m: if(n=1)then return 0:fi: for k from procname(n-1)+1 do m:=k^3: while(length(m)>1)do m:=A007954(m): od: if(m in {0,1,8})then return k: fi: od: end: seq(A116978(n)^3, n=1..50); # Nathaniel Johnston, May 05 2011

fQ[n_] := IntegerQ[ FixedPoint[Times @@ IntegerDigits@# &, n]^(1/3)]; Select[Range[0, 48]^3, fQ@# &] (* Robert G. Wilson v, Apr 03 2006 *)

t(k)=while(k>9, k=prod(i=1, #k=digits(k), k[i])); k
for(n=0, 200, if(ispower(t(n^3), 3), print1(n^3, ", "))); \\ Altug Alkan, Oct 22 2015

a(n) >= A052044(n)^3 for n > 3. - Charles R Greathouse IV, Nov 17 2015

Corrected and extended by Robert G. Wilson v, Apr 03 2006

A333206 a(n) is the least decimal digit of n^3.

Original entry on oeis.org

0, 1, 8, 2, 4, 1, 1, 3, 1, 2, 0, 1, 1, 1, 2, 3, 0, 1, 2, 5, 0, 1, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 2, 3, 0, 2, 4, 0, 2, 1, 0, 1, 0, 0, 1, 1, 3, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 2, 0, 1, 2, 2, 0, 1, 0, 0, 1, 2, 0, 0, 1, 3, 3, 2, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 6, 0, 0, 3, 3, 1, 1

Offset: 0

Robert Israel, Mar 12 2020

Dean Hickerson found an infinite sequence of n such that a(n) > 0 (see Guy, sec F24). Are there infinitely many such that a(n) > 1? If not, what is the greatest n with a(n)=k for each k > 1?

Heuristically, we should expect on the order of ((10-m)^3/100)^d terms n with d digits and a(n) >= m. Since 5^3/100 > 1 > 4^3/100 we should expect infinitely many terms with a(n) >= 5 but only finitely many terms with a(n) >= 6. See A291644 for a(n) = 5. There are only two n <= 10^6 with a(n) >= 6, namely a(2) = 8 and a(92) = 6.

			The least digit of 6^3=216 is 1, so a(6)=1.

R. Guy, Unsolved Problems in Number Theory (Third edition), Springer 2004.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A052044, A054054, A269250, A291639, A291640, A291641, A291642, A291643, A291644.

seq(min(convert(n^3,base,10)),n=0..200);

a(n) = A054054(n^3).

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

cp's OEIS Frontend

A051833 Primes of form (2*10^(5n) - 10^(4n) + 2*10^(3n) + 10^(2n) + 10^n + 1)/3.

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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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A051751 Cubes arising in A051750.

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A052045 Cubes lacking the digit zero in their decimal expansion.

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A052427 Baxter-Hickerson numbers.

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A116978 Cubes whose multiplicative digital root is also a cube.

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A333206 a(n) is the least decimal digit of n^3.

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A051833 Primes of form (210^(5n) - 10^(4n) + 210^(3n) + 10^(2n) + 10^n + 1)/3.

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