A054033 -id:A054033 - cp's OEIS frontend

A054038 Numbers k such that k^2 contains every digit at least once.

32043, 32286, 33144, 35172, 35337, 35757, 35853, 37176, 37905, 38772, 39147, 39336, 40545, 42744, 43902, 44016, 45567, 45624, 46587, 48852, 49314, 49353, 50706, 53976, 54918, 55446, 55524, 55581, 55626, 56532, 57321, 58413, 58455

Offset: 1

Asher Auel, Feb 28 2000

There are 87 terms < 10^5; these are the n such that n^2 uses each digit exactly once. - David Wasserman, Feb 03 2005

The squares in this sequence are in A190682. - Bruno Berselli, May 23 2011

J.-M. De Koninck and A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 239 pp. 39; 178, Ellipses Paris 2004.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 4866 terms from Klaus Brockhaus)

Cf. A016069, A054031, A054032, A054033, A054034, A054035, A054036, A054037, A054039.

IsA054038:=func< n | Seqset(Intseq(n^2)) eq {0,1,2,3,4,5,6,7,8,9} >; [ n: n in [1..60000] | IsA054038(n) ]; // Klaus Brockhaus, May 16 2011

f := []; for i from 0 to 200 do if nops({op(convert(i^2,base,10))})=10 then f := [op(f),i] fi; od; f;

A050278 = Select[FromDigits@#&/@Permutations[Range[0, 9], {10}], # > 10^9 &]; Sqrt[Select[A050278, IntegerQ[Sqrt[#]] &]] (* Alonso del Arte, Jun 18 2011, based on a program by Robert G. Wilson v *)
Select[Sqrt[#]&/@FromDigits/@Select[Permutations[Range[0,9]],#[[1]]>0&], IntegerQ] (* Harvey P. Dale, May 26 2016 *)

is(n)=#vecsort(Vec(Str(n^2)),,8)==10 \\ Charles R Greathouse IV, Jun 18 2011

def ok(n): return len(set(str(n**2))) == 10
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Dec 23 2022

More terms from David Wasserman, Feb 03 2005

A054037 Numbers k such that k^2 contains exactly 9 different digits.

Original entry on oeis.org

10124, 10128, 10136, 10214, 10278, 11826, 12363, 12543, 12582, 12586, 13147, 13268, 13278, 13343, 13434, 13545, 13698, 14098, 14442, 14676, 14743, 14766, 15353, 15681, 15963, 16549, 16854, 17252, 17529, 17778, 17816, 18072, 19023, 19377, 19569, 19629, 20089

Offset: 1

Asher Auel, Feb 28 2000

There are three prime numbers {13147, 20089, 21397} and corresponding squares {172843609, 403567921, 457831609} necessarily contain zero (otherwise n and n^2 are divisible by 3). - Zak Seidov, Jan 18 2012

Sean A. Irvine, Table of n, a(n) for n = 1..10000 (terms 1..83 from Zak Seidov)

Cf. A016069, A054031, A054032, A054033, A054034, A054035, A054036, A054038, A054039, A156977.

f := []; for i from 0 to 200 do if nops({op(convert(i^2,base,10))})=9 then f := [op(f),i] fi; od; f;

okQ[n_] := Module[{n2=n^2}, Max[DigitCount[n2,10]]==1 && IntegerLength[n2]==9]; Select[Range[20000], okQ]  (* Harvey P. Dale, Mar 20 2011 *)

from itertools import count, islice
def agen(): yield from (k for k in count(10**4) if len(set(str(k*k)))==9)
print(list(islice(agen(), 37))) # Michael S. Branicky, May 24 2022

A054039 a(n)^2 is the least square to contain n different decimal digits.

Original entry on oeis.org

0, 4, 13, 32, 113, 322, 1017, 3206, 10124, 32043

Offset: 1

Asher Auel, Feb 28 2000

It turns out that "...at least n..." and "...exactly n..." yield the same (and thus strictly increasing) sequence. - M. F. Hasler, Feb 02 2009

			13^2=169 is the first square to contain exactly 3 different digits; 322^2=103684 is the first square to contain exactly 6 different digits.

Cf. A016069, A054031, A054032, A054033, A054034, A054035, A054036, A054037, A054038.

A054039(n,k=0) = { while( #Set(Vec(Str(k^2)))M. F. Hasler, Feb 02 2009 */

Minor rewording, added comment, keywords "easy,full" and PARI code M. F. Hasler, Feb 02 2009

A054031 Numbers whose square contains exactly 3 distinct digits.

Original entry on oeis.org

13, 14, 16, 17, 18, 19, 23, 24, 25, 27, 28, 29, 31, 34, 35, 39, 40, 41, 45, 46, 47, 50, 56, 58, 60, 62, 63, 65, 67, 68, 70, 75, 76, 77, 80, 81, 83, 85, 90, 91, 92, 94, 97, 101, 102, 107, 108, 110, 111, 119, 120, 121, 122, 129, 131, 141, 149, 150, 162, 165

Offset: 1

Asher Auel, Feb 29 2000

Giovanni Resta, Table of n, a(n) for n = 1..12299 (terms < 10^18, first 1000 terms from T. D. Noe)
Michael Geißer, Theresa Körner, Sascha Kurz, and Anne Zahn, Squares with three digits, arXiv:2112.00444 [math.NT], 2021-2022.
Carlos Rivera, Puzzle 313. Squares having only k distinct digits, The Prime Puzzles and Problems Connection.

Cf. A235718, A016069, A054032, A054033, A054034, A054035, A054036, A054037, A054038, A054039.

f := []; for i from 0 to 200 do if nops({op(convert(i^2,base,10))})=3 then f := [op(f),i] fi; od; f;

t = {}; n = -1; While[Length[t] < 50, n++; If[Length[Union[IntegerDigits[n^2]]] == 3, AppendTo[t, n]]] (* T. D. Noe, Apr 26 2013 *)
Select[Range[200],Length[Union[IntegerDigits[#^2]]]==3&] (* Harvey P. Dale, Aug 17 2014 *)

is(n)=#Set(digits(n^2))==3 \\ Charles R Greathouse IV, Feb 11 2017

A235718(n) = a(n)^2. - Giovanni Resta, Apr 28 2017

A054032 Numbers n such that n^2 contains exactly 4 different digits.

Original entry on oeis.org

32, 33, 36, 37, 42, 43, 44, 48, 49, 51, 52, 53, 54, 55, 57, 59, 61, 64, 66, 69, 71, 72, 73, 74, 78, 79, 82, 84, 86, 87, 89, 93, 95, 96, 98, 99, 103, 104, 105, 106, 112, 114, 115, 123, 125, 127, 130, 132, 135, 138, 139, 140, 143, 145, 146, 151, 155, 156, 157, 158

Offset: 1

Asher Auel, Feb 29 2000

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A016069, A054031, A054033, A054034, A054035, A054036, A054037, A054038, A054039.

f := []; for i from 0 to 200 do if nops({op(convert(i^2,base,10))})=4 then f := [op(f),i] fi; od; f;

t = {}; n = -1; While[Length[t] < 50, n++; If[Length[Union[IntegerDigits[n^2]]] == 4, AppendTo[t, n]]] (* T. D. Noe, Apr 26 2013 *)

is(n)=#Set(digits(n^2))==4 \\ Charles R Greathouse IV, Feb 11 2017

A054034 Numbers n such that n^2 contains exactly 6 different digits.

Original entry on oeis.org

322, 323, 324, 328, 352, 353, 364, 367, 374, 375, 397, 403, 405, 413, 416, 425, 442, 445, 456, 458, 463, 487, 504, 507, 508, 509, 529, 542, 557, 564, 567, 571, 572, 574, 584, 589, 591, 593, 597, 598, 616, 618, 621, 625, 626, 629, 634, 637, 639, 645, 647

Offset: 1

Asher Auel, Feb 29 2000

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A016069, A054031, A054032, A054033, A054035, A054036, A054037, A054038, A054039.

f := []; for i from 0 to 200 do if nops({op(convert(i^2,base,10))})=6 then f := [op(f),i] fi; od; f;

Select[Range[700],Count[DigitCount[#^2],0]==4&] (* Harvey P. Dale, May 10 2021 *)

A054035 Numbers n such that n^2 contains exactly 7 different digits.

Original entry on oeis.org

1017, 1023, 1024, 1027, 1028, 1036, 1037, 1042, 1113, 1117, 1164, 1175, 1176, 1197, 1228, 1267, 1268, 1277, 1302, 1307, 1323, 1328, 1343, 1352, 1355, 1375, 1395, 1405, 1428, 1433, 1441, 1442, 1444, 1463, 1541, 1593, 1594, 1628, 1646, 1648, 1701, 1706

Offset: 1

Asher Auel, Feb 29 2000

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A016069, A054031, A054032, A054033, A054034, A054036, A054037, A054038, A054039.

f := []; for i from 0 to 200 do if nops({op(convert(i^2,base,10))})=7 then f := [op(f),i] fi; od; f;

Select[Range[2000],Count[DigitCount[#^2],0]==3&] (* Harvey P. Dale, Jul 28 2012 *)

A054036 Numbers n such that n^2 contains exactly 8 different digits.

Original entry on oeis.org

3206, 3267, 3268, 3292, 3674, 3678, 3698, 3723, 3734, 4047, 4097, 4157, 4175, 4455, 4537, 4554, 4616, 4634, 4663, 4804, 4814, 4896, 4913, 4967, 4987, 5376, 5529, 5699, 5742, 5853, 5899, 5904, 5905, 5968, 6043, 6071, 6095, 6098, 6127, 6176, 6181, 6199

Offset: 1

Asher Auel, Feb 28 2000

			3206 is in the sequence because 3206^2 = 10278436 and 10278436 contains exactly eight different digits: 0, 1, 2, 3, 4, 6, 7 and 8.

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A016069, A054031, A054032, A054033, A054034, A054035, A054037, A054038, A054039, A235723.

f := []; for i from 0 to 200 do if nops({op(convert(i^2,base,10))})=8 then f := [op(f),i] fi; od; f;

Select[Range[7000],Count[DigitCount[#^2],0]==2&] (* Harvey P. Dale, Aug 10 2017 *)

A235720 Squares which have one or more occurrences of exactly five different digits.

Original entry on oeis.org

12769, 13456, 13689, 13924, 15376, 15876, 16384, 17689, 17956, 18496, 18769, 20164, 20736, 21609, 21904, 23104, 23409, 23716, 28561, 29584, 30276, 30625, 30976, 31684, 32041, 32761, 34596, 35721, 36481, 37249, 38025, 38416, 39204, 39601, 41209, 43681, 45369

Offset: 1

Colin Barker, Jan 15 2014

The first term having a repeated digit is 100489.

			30976 is in the sequence because 30976 = 176^2 and 30976 contains exactly five different digits: 0, 3, 6, 7 and 9.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A235717-A235719, A235721-A235724, A225218.

s=[]; for(n=1, 1200, if(#vecsort(eval(Vec(Str(n^2))),,8)==5, s=concat(s, n^2))); s

a(n) = A054033(n)^2.

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A054038 Numbers k such that k^2 contains every digit at least once.

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A054037 Numbers k such that k^2 contains exactly 9 different digits.

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A054039 a(n)^2 is the least square to contain n different decimal digits.

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A054031 Numbers whose square contains exactly 3 distinct digits.

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A054032 Numbers n such that n^2 contains exactly 4 different digits.

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A054034 Numbers n such that n^2 contains exactly 6 different digits.

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A054035 Numbers n such that n^2 contains exactly 7 different digits.

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A054036 Numbers n such that n^2 contains exactly 8 different digits.

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