A052041 -id:A052041 - cp's OEIS frontend

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

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

A104315 Numbers having in decimal representation at least one zero, but with no zero in their square.

Original entry on oeis.org

106, 107, 108, 109, 204, 206, 207, 208, 209, 304, 306, 307, 308, 309, 404, 406, 407, 408, 409, 604, 606, 607, 608, 704, 706, 707, 804, 806, 807, 808, 809, 904, 907, 908, 909, 1056, 1057, 1058, 1059, 1061, 1062, 1063, 1065, 1066, 1067, 1069, 1072, 1073

Offset: 1

Reinhard Zumkeller, Mar 01 2005

A104316(n) = a(n)^2.

			909^2 = 826281, therefore 909 is a term.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A104317, A052041.

a104315 n = a104315_list !! (n-1)
a104315_list = filter (\x -> a168046 x == 0 && a168046 (x ^ 2) == 1) [1..]
-- Reinhard Zumkeller, Jan 03 2014

isok(n) = !vecmin(digits(n)) && vecmin(digits(n^2)); \\ Michel Marcus, Jan 03 2014

(1 - A168046(a(n))) * A168046(A000290(a(n))) = 1. - Reinhard Zumkeller, Jan 03 2014

A069557 Squares in which the k-th significant digit either divides k or is a multiple of k. Is 1 only in case k has no other single-digit divisor.

Original entry on oeis.org

1, 4, 9, 25, 49, 64, 81, 324, 361, 625, 961, 4624, 8649, 7354944

Offset: 1

Amarnath Murthy, Mar 22 2002

If the smallest prime divisor of n is > 7 only in such case the n-th digit is 1.
a(15) > 10^25 if it exists. - David A. Corneth, May 05 2024
The definition implies that no digit can be zero. - N. J. A. Sloane, May 05 2024

			8649 is a member in which the fourth digit is 8, a multiple of 4, the third one is 6, a multiple of 3, the second one is 4, a multiple of 2 and the least significant digit is 9.

David A. Corneth, PARI program

Cf. A052041, A069556, A069560.

okQ[n_]:=Module[{idn=IntegerDigits[n]}, Count[idn,0]==0 && And@@Divisible[idn,Range[Length[idn],1,-1]]]; Select[Range[500]^2, okQ]  (* Harvey P. Dale, Dec 20 2010 *)

PARI
```
\\ See PARI link
```

More terms from Sascha Kurz, Mar 23 2002

Offset corrected and a(14) from Sean A. Irvine, May 04 2024

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

A104316 A104315(n)^2.

Original entry on oeis.org

11236, 11449, 11664, 11881, 41616, 42436, 42849, 43264, 43681, 92416, 93636, 94249, 94864, 95481, 163216, 164836, 165649, 166464, 167281, 364816, 367236, 368449, 369664, 495616, 498436, 499849, 646416, 649636, 651249, 652864, 654481

Offset: 1

Reinhard Zumkeller, Mar 01 2005

Subsequence of A052041, squares having in their decimal representation no zeros.

Cf. A104317.

A118489 Squares for which the product of the digits is a triangular number.

Original entry on oeis.org

0, 1, 16, 25, 49, 100, 400, 576, 900, 1024, 1089, 1521, 1600, 2025, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 3025, 3600, 4096, 4900, 5041, 5625, 6084, 6400, 7056, 8100, 9025, 9409, 9604, 9801, 10000, 10201, 10404, 10609, 10816, 11025, 11236, 12100, 14400

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), May 05 2006

Includes all squares not in A052041. - Robert Israel, May 25 2020

			26569 is in the sequence because (1) it is a square, (2) the product of its digits is 2*6*5*6*9=3240 which is a triangular number.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000217, A052041.

filter:= n -> issqr(1+8*convert(convert(n,base,10),`*`)):
select(filter, [seq(i^2,i=1..1000)]); # Robert Israel, May 25 2020

Select[Range[0,120]^2,OddQ[Sqrt[8 (Times@@IntegerDigits[#]) +1]]&] (* Harvey P. Dale, Jul 06 2021 *)

1600 inserted by Robert Israel, May 25 2020

A257649 Squares that are the concatenation of two integers (without leading zeros) the sum of which is also a square.

Original entry on oeis.org

36, 81, 169, 196, 324, 361, 576, 729, 841, 1156, 1521, 1681, 1764, 2809, 3249, 3481, 4356, 5625, 6084, 6241, 6724, 7396, 7569, 7744, 7921, 8281, 9216, 12321, 12544, 12769, 12996, 13689, 15129, 16384, 17424, 18769, 19881, 24964, 25600, 31684, 32041, 34596, 36864, 38416, 39601

Offset: 1

Reiner Moewald, Jul 25 2015

Squares that can be split up in more than one way, e.g., 729 (72 + 9 and 7 + 29), appear only once.
The number of such squares is infinite, since 39...960...01 (the numbers of the digits 9 and 0 is equal) can be split up into 3 and 9...960..01 with 3 + 9...960...01 = (100...0-2)^2 and 39...960...01 = (2*100...0 - 1)^2.
From Robert G. Wilson v, Aug 06 2015: (Start)
Number of terms < 10^k: 0, 2, 9, 27, 66, 149, 370, 910, 2164, 5325, 12916, 29448, ..., .
Terms which are members of A257649 in more than one way: 729, 7569, 15129, 56169, 86436, 123201, ..., .
Terms which are members of A257649 in more than two way: 881377344, 3784833441, 39999600001, 54444755556, 71111288889, 89999400001, 159999200001, 321111488889, 751111688889, ..., .
Least term which is a member of A257649 in k ways: 36, 729, 881377344, 399999960000001, ..., . (End)

			36 = 6^2 and 3 + 6 = 9 = 3^2.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 909 terms from Reiner Moewald)

Subsequence of A052041.

f[n_] := Block[{a, b, c, k = 1, idn = IntegerDigits@ n, lng, lst = {}}, lng = Length@ idn; While[k < lng, a = FromDigits[ Take[idn, {1, k}]]; b = FromDigits[ Take[idn, {k + 1, lng}]]; c = a*10^(lng - k) + b; If[b > 0 && Floor[1 + Log10@ b] == lng - k && IntegerQ@ Sqrt[a + b], AppendTo[lst, c]]; k++]; Length@ lst]; k = 1; lst = {}; While[k < 201, If[ f[k^2] > 0, AppendTo[lst, k^2]]; k++]; lst (* Robert G. Wilson v, Aug 06 2015 *)
ctiQ[n_]:=AnyTrue[Total/@Select[Table[FromDigits/@TakeDrop[IntegerDigits[n],d],{d,IntegerLength[ n]-1}],IntegerLength[#[[1]]]+IntegerLength[#[[2]]] ==IntegerLength[ n]&],IntegerQ[ Sqrt[#]]&]; Select[Range[200]^2,ctiQ] (* Harvey P. Dale, Jun 04 2023 *)

import math
print("Start")
list =[]
for i in range(1,1000):
   a = i*i
   b = str(a)
   l = len(b)
   for j in range(1, l):
      a_1 = b[:j]
      a_2 = b[j:]
      c = int(a_1)+int(a_2)
      sqrt_c = int(math.sqrt(int(c)))
      if (sqrt_c * sqrt_c == c) and (int(a_2[:1]) > 0):
         if not a in list:
            list.append(a)
print(list)
print("End")

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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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A052045 Cubes lacking the digit zero in their decimal expansion.

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A104315 Numbers having in decimal representation at least one zero, but with no zero in their square.

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A069557 Squares in which the k-th significant digit either divides k or is a multiple of k. Is 1 only in case k has no other single-digit divisor.

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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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Maple

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