A052040 -id:A052040 - cp's OEIS frontend

A052045 Cubes lacking the digit zero in their decimal expansion.

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

A255398 Numbers k such that k^2 lacks the digit 1 in its decimal expansion.

Original entry on oeis.org

0, 2, 3, 5, 6, 7, 8, 15, 16, 17, 18, 20, 22, 23, 24, 25, 26, 27, 28, 30, 45, 47, 48, 50, 52, 53, 55, 57, 58, 60, 62, 63, 64, 65, 66, 67, 68, 70, 73, 74, 75, 76, 77, 78, 80, 82, 83, 84, 85, 86, 87, 88, 92, 93, 94, 95, 97, 98, 143, 144, 150, 153, 155, 156, 157, 158

Offset: 1

Vincenzo Librandi, Feb 22 2015

			98 is in this sequence because 98^2 = 9604.
99 is not in this sequence because 99^2 = 9801.

Mohammed Yaseen, Table of n, a(n) for n = 1..10000

Cf. A052383, A052040, A099150.

[n: n in [0..200] | not 1 in Intseq(n^2)];

filter:= n -> not member(1, convert(n^2,base,10)):
select(filter, [$0..200]); # Robert Israel, Apr 27 2023

Select[Range[0, 200], DigitCount[#^2, 10, 1]==0 &]

isok(k) = !vecsearch(Set(digits(k^2)), 1); \\ Michel Marcus, Apr 29 2023

def ok(k): return "1" not in str(k**2)
print([k for k in range(160) if ok(k)]) # Michael S. Branicky, Apr 27 2023

From Mohammed Yaseen, Apr 18 2023: (Start)
The smallest n-digit term ~ sqrt(2) * 10^(n-1).
The largest n-digit term = 10^n - 2 (see A099150). (End)

A257760 Zeroless numbers n such that the products of the decimal digits of n and n^2 coincide.

Original entry on oeis.org

1, 1488, 3381, 14889, 18489, 181965, 262989, 338646, 358489, 367589, 437189, 438329, 479285, 781839, 964941, 1456589, 1763954, 2579285, 2868489, 3365285, 3419389, 3451988, 3584889, 3625619, 4378829, 4653989, 6868877, 7295986, 9548479, 14529839, 14534488

Offset: 1

Pieter Post, May 07 2015

It is unknown if this sequence is infinite.
Number of terms < 10^n: 1, 1, 1, 3, 5, 15, 29, 75, 211, 583, 1694, ..., . - Robert G. Wilson v, May 25 2015
Also nontrivial numbers n such that the products of the decimal digits of n and n^2 are equal. Trivial solutions are any number which contains a zero in its decimal expansion. - Robert G. Wilson v, May 11 2015

			1488 is in the sequence since 1488^2 = 2214144 and we have 256 = 1*4*8*8 = 2*2*1*4*1*4*4.
3381 is in the sequence because 3381^2 = 11431161 and 72 = 3*3*8*1 = 1*1*4*3*1*1*6*1.

Robert G. Wilson v, Table of n, a(n) for n = 1..1747 (first 75 terms from Pieter Post)

Subsequence of A052040.

Cf. A000290, A007954, A029793, A052382, A257763.

fQ[n_] := Times @@ IntegerDigits[n] == Times @@ IntegerDigits[n^2] > 0; Select[ Range@ 10000000, fQ] (* Robert G. Wilson v, May 07 2015 *)

isok(n) = (d = digits(n)) && vecmin(d) && (dd = digits(n^2)) && (prod(k=1, #d, d[k]) == prod(k=1, #dd, dd[k])); \\ Michel Marcus, May 07 2015

A328780 Nonnegative integers k such that k and k^2 have the same number of nonzero digits.

Original entry on oeis.org

0, 1, 2, 3, 10, 20, 30, 100, 200, 245, 247, 249, 251, 253, 283, 300, 448, 548, 949, 1000, 1249, 1253, 1416, 1747, 1749, 1751, 1753, 1755, 2000, 2245, 2247, 2249, 2251, 2253, 2429, 2450, 2451, 2470, 2490, 2498, 2510, 2530, 2647, 2830, 3000, 3747, 3751, 4480, 4899

Offset: 1

Bernard Schott, Oct 27 2019

The idea of this sequence comes from the 1st problem of the 28th British Mathematical Olympiad in 1992 (see the link).
This sequence is infinite because the family of integers {10^k, k >= 0} (A011557) belongs to this sequence.
The numbers m, m + 1, m + 2 where m = 49*10^k - 3, or m = 99*10^k - 3, k >= 3 are terms with all nonzero digits. - Marius A. Burtea, Dec 21 2020

			247^2 = 61009, hence 247 and 61009 both have 3 nonzero digits, 247 is a term.

A. Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, Pb 1 pp. 57 and 109 (1992)

Giovanni Resta, Table of n, a(n) for n = 1..10000.
British Mathematical Olympiad, 1992 - Problem 1.
Index to sequences related to Olympiads.

Subsequences: A011557, A093136, A093138.
Cf. A052040, A104315, A134844.
Cf. A328781, A328782, A328783.

nz:=func; [k:k in [0..5000] | nz(k) eq nz(k^2)]; // Marius A. Burtea, Dec 21 2020

q:= n->(f->f(n)=f(n^2))(t->nops(subs(0=[][], convert(t, base, 10)))):
select(q, [$0..5000])[];  # Alois P. Heinz, Oct 27 2019

Select[Range[0, 5000], Equal @@ Total /@ Sign@ IntegerDigits[{#, #^2}] &] (* Giovanni Resta, Feb 27 2020 *)

isok(k) = hammingweight(digits(k)) == hammingweight(digits(k^2)); \\ Michel Marcus, Dec 22 2020

More terms from Alois P. Heinz, Oct 27 2019

A328781 Nonnegative integers k such that k and k^2 contain the same number of zero digits in their decimal expansion.

Original entry on oeis.org

0, 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, 104, 105

Offset: 1

Bernard Schott, Oct 27 2019

Inspired by A328780.
This sequence is not a duplicate of A052040. The first 72 terms until 96 are exactly the same but a(73) = 104 belongs to this sequence because 104^2 = 10816, but 104 doesn't belong to A052040 because there is one zero digit in the decimal expansion of 104^2.
The nonnegative integers that do not belong to this sequence are divided into three sequences:
1) A104315 = A052040 \ {this sequence}: Numbers k such that k contains at least one zero, but k^2 contains no zero (e.g., 106 with 106^2 = 11236).
2) A134844 = Numbers k such that k contains no zero but k^2 contains at least one zero (e.g., 32 with 32^2 = 1024).
3) A328783 = Numbers k such that k and k^2 contain at least one zero but not the same number of zeros (e.g., 101 with 101^2 = 10201).
Another sequence is A328782 = {this sequence} \ A052040 which lists the positive integers that have the same positive number of zeros in their decimal expansions as in their squares. The first two examples > 0 are 104 with 104^2 = 10816 and 105 with 105^2 = 11025.

			12 and 144 = 12^2 have no digit zero in their decimal representation, so 12 is a term.
203 and 41209 = 203^2 both have one digit zero in their decimal representation, so 203 is also a term.

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

Cf. A052040, A104315, A134844.

Cf. A323780, A328782, A328783.

select(t -> numboccur(0, convert(t^2,base,10))=numboccur(0, convert(t,base,10)), [$0..200]); # Robert Israel, Oct 27 2019

Select[Range[0, 105], Equal @@ Total /@ (1 - Sign@ IntegerDigits[{#, #^2}]) &] (* Giovanni Resta, Feb 27 2020 *)

A328782 Integers k such that k and k^2 contain the same number > 0 of digits zero in their decimal expansion.

Original entry on oeis.org

0, 104, 105, 203, 205, 302, 303, 305, 402, 403, 405, 504, 505, 506, 507, 508, 509, 601, 602, 603, 605, 609, 701, 702, 703, 705, 708, 709, 801, 802, 803, 805, 901, 902, 903, 905, 906, 1006, 1007, 1008, 1009, 1011, 1012, 1013, 1014, 1016, 1017, 1018, 1019, 1021

Offset: 1

Bernard Schott, Oct 28 2019

			703 and 494209 = 703^2 both have one zero digit in their decimal expansion.

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

Equals A328781 \ A052040.

Cf. A104315, A134844, A328780, A328783.

f:= n-> numboccur(0, convert(n, base, 10)):
q:= n-> ((x, y)-> x>0 and x=y)(f(n), f(n^2)):
select(q, [$0..1030])[];  # Alois P. Heinz, Oct 28 2019

Select[Range[0, 1100], DigitCount[#, 10, 0] == DigitCount[#^2, 10, 0] > 0 &] (* Giovanni Resta, Feb 27 2020 *)

More terms from Alois P. Heinz, Oct 28 2019

A328783 Numbers k such that k and k^2 contain at least one zero but not the same number of 0's.

Original entry on oeis.org

10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 101, 102, 103, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 201, 202, 210, 220, 230, 240, 250, 260, 270, 280, 290, 300, 301, 310, 320, 330, 340, 350, 360, 370, 380, 390, 400, 401, 410, 420, 430, 440, 450, 460, 470

Offset: 1

Bernard Schott, Oct 28 2019

This sequence is one of the three sequences whose numbers k and k^2 don't contain the same number of 0, the two others are A104315 and A134844.

			201 and 40401 = 201^2 have both at least one zero but not the same number of 0 in their decimal expansion, hence, 201 is a term.

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

Cf. A052040, A104315, A134844, A328780, A328781, A328782.

f:= n-> numboccur(0, convert(n, base, 10)):
q:= n-> ((x, y)-> x>0 and y>0 and x<>y)(f(n), f(n^2)):
select(q, [$0..500])[];  # Alois P. Heinz, Oct 28 2019

Select[Range[0, 470], (x = DigitCount[#, 10, 0]) > 0 && (y = DigitCount[ #^2, 10, 0]) > 0 && x != y &] (* Giovanni Resta, Feb 27 2020 *)

More terms from Alois P. Heinz, Oct 28 2019

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

A376003 Positive integers k such that each digit of k^2 is a factor of k.

Original entry on oeis.org

1, 6, 12, 36, 54, 108, 156, 168, 192, 204, 288, 306, 408, 432, 486, 696, 804, 1104, 1146, 1188, 1488, 1512, 1632, 1764, 1806, 1932, 2232, 2904, 3114, 3408, 3456, 3528, 4014, 4104, 4392, 4596, 4608, 4704, 4788, 4872, 4932, 4944, 5208, 5304, 5868, 6012, 6696, 6792

Offset: 1

Sam N. Harrison, Sep 28 2024

0 is never a factor so k^2 must be zeroless and this sequence is a subset of A052040.
The first term > 1 that is not divisible by 6 is 47768.
From Andrew Howroyd, Sep 28 2024: (Start)
Except for the first term, all terms are even since all squares with at least 2 digits contain an even digit. This implies k^2 cannot contain the digit 5.
All numbers of the form (100*1000^k-1)/3+3 are terms. These are the numbers 36, 33336, 33333336, 33333333336, etc. This shows that the sequence is infinite. (End)

			k = 12 is a term since k^2 = 144 has digits 1 and 4 and both are factors of k.
k = 2 is not a term since k^2 = 4 has a digit 4 which is not a factor of k.

Cf. A052040, A034838.

q:= n-> andmap(x-> x>0 and irem(n, x)=0, convert(n^2, base, 10)):
select(q, [$1..10000])[];  # Alois P. Heinz, Sep 28 2024

isok(k) = my(d=Set(digits(k^2))); if(!vecmin(d), return(0)); for (i=1, #d, if (k % d[i], return(0))); return(1); \\ Michel Marcus, Sep 28 2024

def is_valid_k(k):
    k_squared = k ** 2
    for digit in str(k_squared):
        d = int(digit)
        if d == 0 or k % d != 0:
            return False
    return True
def find_valid_k(max_k):
    valid_k = []
    for k in range(1, max_k + 1):
        if is_valid_k(k):
            valid_k.append(k)
    return valid_k
max_k = 10000
result = find_valid_k(max_k)
print(result)

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A052045 Cubes lacking the digit zero in their decimal expansion.

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A255398 Numbers k such that k^2 lacks the digit 1 in its decimal expansion.

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A257760 Zeroless numbers n such that the products of the decimal digits of n and n^2 coincide.

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A328780 Nonnegative integers k such that k and k^2 have the same number of nonzero digits.

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A328781 Nonnegative integers k such that k and k^2 contain the same number of zero digits in their decimal expansion.

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A328782 Integers k such that k and k^2 contain the same number > 0 of digits zero in their decimal expansion.

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A328783 Numbers k such that k and k^2 contain at least one zero but not the same number of 0's.

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