A054038 -id:A054038 - cp's OEIS frontend

A085545 Smaller of number pairs from A054038 which together use all ten digits once.

35172, 57321, 58413, 59403

Offset: 1

Lekraj Beedassy, Jul 03 2003

			57321 in A054038 has the companion 60984 and together use all ten digits once.

S. S. Gupta, Fascinating Squares: Recent Study: Curious Observations

Cf. A054038.

Corrected by David Wasserman, Feb 03 2005

A029793 Numbers k such that k and k^2 have the same set of digits.

Original entry on oeis.org

0, 1, 10, 100, 1000, 4762, 4832, 10000, 10376, 10493, 11205, 12385, 12650, 14829, 22450, 23506, 24605, 26394, 34196, 36215, 47620, 48302, 48320, 49827, 64510, 68474, 71205, 72510, 72576, 74510, 74528, 79286, 79603, 79836, 94583, 94867, 96123, 98376

Offset: 1

Patrick De Geest

This sequence has density 1: almost all numbers k have all 10 digits in both k and k^2. - Franklin T. Adams-Watters, Jun 28 2011

			{0, 1, 3, 4, 9} = digits of a(10) = 10493 and of 10493^2 = 110103049;
{0, 1, 2, 5, 6} = digits of a(100) = 162025 and of 162025^2 = 26252100625;
{0, 1, 3, 4, 6, 7, 8} = digits of a(1000) = 1764380 and of 1764380^2 = 3113036784400;
{1, 2, 3, 4, 7, 8, 9} = digits of a(10000) = 14872239 and of 14872239^2 = 221183492873121.

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

Cf. A029795, A029797, A030091; A178501 is a subsequence, A054038, A171102, A232659-A232662.

import Data.List (nub, sort)
a029793 n = a029793_list !! (n-1)
a029793_list = filter (\x -> digs x == digs (x^2)) [0..]
   where digs = sort . nub . show
-- Reinhard Zumkeller, Jun 27 2011

[ n: n in [0..10^5] | Set(Intseq(n)) eq Set(Intseq(n^2)) ];  // Bruno Berselli, Jun 28 2011

seq(`if`(convert(convert(n,base,10),set) = convert(convert(n^2,base,10),set), n, NULL), n=0..100000); # Nathaniel Johnston, Jun 28 2011

digitSet[n_] := Union[IntegerDigits[n]]; Select[Range[0, 99000], digitSet[#] == digitSet[#^2] &] (* Jayanta Basu, Jun 02 2013 *)

isA029793(n)=Set(Vec(Str(n)))==Set(Vec(Str(n^2))) \\ Charles R Greathouse IV, Jun 28 2011

(0L to 99999L).filter(n => n.toString.toCharArray.toSet == (n * n).toString.toCharArray.toSet) // Alonso del Arte, Jan 19 2020

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)).

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

A225218 Square numbers containing all the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

Original entry on oeis.org

1026753849, 1042385796, 1098524736, 1237069584, 1248703569, 1278563049, 1285437609, 1382054976, 1436789025, 1503267984, 1532487609, 1547320896, 1643897025, 1827049536, 1927385604, 1937408256, 2076351489, 2081549376, 2170348569, 2386517904, 2431870596

Offset: 1

Reiner Moewald, May 02 2013

The first term having a repeated digit is 10057482369. - Colin Barker, Jan 15 2014

			1026753849 is in the sequence because 1026753849 = 32043^2 and 1026753849 contains all ten digits 0, ..., 9.

Reiner Moewald, Table of n, a(n) for n = 1..4865

Supersequence of A036745.

Cf. A235717-A235724.

Select[#^2 &[Range[1000000]], Length[Union[IntegerDigits[#]]] == 10 &] (* Geoffrey Critzer, Jan 04 2015 *)

s=[]; for(n=1, 100000, if(#vecsort(eval(Vec(Str(n^2))), , 8)==10, s=concat(s, n^2))); s \\ Colin Barker, Jan 15 2014

from itertools import count, islice
def c(n): return len(set(str(n))) == 10
def agen(): yield from (k*k for k in count(31622) if c(k*k))
print(list(islice(agen(), 21))) # Michael S. Branicky, Dec 27 2022

a(n) = A054038(n)^2. - Colin Barker, Jan 15 2014

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

cp's OEIS Frontend

A085545 Smaller of number pairs from A054038 which together use all ten digits once.

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A029793 Numbers k such that k and k^2 have the same set of digits.

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

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

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A225218 Square numbers containing all the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

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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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