A054037 -id:A054037 - 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

A156977 Numbers n such that n^2 contains every decimal digit exactly once.

Original entry on oeis.org

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, 58554, 59403, 60984

Offset: 1

Zak Seidov, Feb 20 2009

There are exactly 87 such numbers, none of them being prime.

Since 0 + 1 +...+ 9 = 5*9, every pandigital number is divisible by 9, hence every term of this sequence is divisible by 3 and so cannot be a prime. - Giovanni Resta, Mar 19 2013 [Comment expanded by N. J. A. Sloane, Jan 15 2022]

Giovanni Resta, Table of n, a(n) for n = 1..87 (full sequence)
S. C. Gould, Question 15734, The Educational Times, and Journal of the College of Preceptors 58 (1905), nr. 527 (March 1), p. 157; Solution 15734, Ibid., nr. 529 (May 1), p. 235.

Cf. A036745, A054037, A054038.

[n: n in [Floor(Sqrt(1023456789))..Ceiling(Sqrt(9876543210))] | Set(Intseq(n^2)) eq {0..9}]; // Bruno Berselli, Mar 19 2013 (after Giovanni Resta)

lim:=floor(sqrt(9876543210)): for n from floor(sqrt(1023456789)) to lim do d:=[op(convert(n^2, base, 10))]: pandig:=true: for k from 0 to 9 do if(numboccur(k, d)<>1)then pandig:=false: break: fi: od: if(pandig)then printf("%d, ",n): fi: od: # Nathaniel Johnston, Jun 22 2011

Select[Range[Floor@Sqrt@1023456789, Ceiling@Sqrt@9876543210], Sort@IntegerDigits[#^2] == Range[0, 9] &] (* Giovanni Resta, Mar 19 2013 *)
Select[Range[31992,99381,3],Union[DigitCount[#^2]]=={1}&] (* Harvey P. Dale, Jan 17 2022 *)

a(n) = sqrt(A036745(n)).

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

A071519 Numbers whose square is a zeroless pandigital number (i.e., use the digits 1 through 9 once).

Original entry on oeis.org

11826, 12363, 12543, 14676, 15681, 15963, 18072, 19023, 19377, 19569, 19629, 20316, 22887, 23019, 23178, 23439, 24237, 24276, 24441, 24807, 25059, 25572, 25941, 26409, 26733, 27129, 27273, 29034, 29106, 30384

Offset: 1

Lekraj Beedassy, Jun 20 2002

A subset of A054037.

Cf. A036744, A054038, A156977.

lim:=floor(sqrt(987654321)): for n from floor(sqrt(123456789)) to lim do d:=[op(convert(n^2, base, 10))]: pandig:=true: for k from 1 to 9 do if(numboccur(k, d)<>1)then pandig:=false: break: fi: od: if(pandig)then printf("%d, ", n): fi: od: # Nathaniel Johnston, Jun 22 2011

Sqrt[#]&/@Select[FromDigits/@Permutations[Range[9]],IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, Sep 23 2011 *)
Select[Range[11112, 31427,3], DigitCount[#^2] == {1,1,1,1,1,1,1,1,1,0} &]  (* Zak Seidov, Jan 11 2012 *)

A071519 = select( {is_A071519(n,L=[1..9])=vecsort(digits(n^2))==L}, [1e5\9..1e5\3]) \\ M. F. Hasler, Jun 28 2023

a(n) = sqrt(A036744(n)). - Zak Seidov, Jan 11 2012

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

A054033 Numbers n such that n^2 contains exactly 5 different digits.

Original entry on oeis.org

113, 116, 117, 118, 124, 126, 128, 133, 134, 136, 137, 142, 144, 147, 148, 152, 153, 154, 169, 172, 174, 175, 176, 178, 179, 181, 186, 189, 191, 193, 195, 196, 198, 199, 203, 209, 213, 214, 217, 219, 224, 226, 228, 232, 233, 248, 252, 259, 267, 268, 269

Offset: 1

Asher Auel, Feb 29 2000

The first 66 terms are the only ones whose squares contain no repeated digits. - Charles R Greathouse IV, Feb 09 2015

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A016069, A054031, A054032, A054034, A054035, A054036, A054037, A054038, A054039.

f := []; for i from 0 to 200 do if nops({op(convert(i^2,base,10))})=5 then f := [op(f),i] fi; od; f;

Select[Range[101,500],Count[DigitCount[#^2],0]==5&] (* Harvey P. Dale, Feb 08 2015 *)

is(n)=#Set(digits(n^2))==5 \\ Charles R Greathouse IV, Feb 08 2015

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 *)

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

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A156977 Numbers n such that n^2 contains every decimal digit exactly once.

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

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A071519 Numbers whose square is a zeroless pandigital number (i.e., use the digits 1 through 9 once).

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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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A054033 Numbers n such that n^2 contains exactly 5 different digits.

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