A045540 -id:A045540 - cp's OEIS frontend

A052046 Squares whose digits occur with the same frequency.

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 169, 196, 256, 289, 324, 361, 529, 576, 625, 729, 784, 841, 961, 1024, 1089, 1296, 1369, 1764, 1849, 1936, 2304, 2401, 2601, 2704, 2809, 2916, 3025, 3249, 3481, 3721, 4096, 4356, 4761, 5041, 5184, 5329, 5476, 6084, 6241

Offset: 1

Patrick De Geest, Dec 15 1999

Contains all members of A078255 that are < 10^10. 7744 is the first member that is not a member of A078255. - David Wasserman, Jun 27 2006

Robert Israel, Table of n, a(n) for n = 1..10000
P. De Geest, Numbers whose digits occur with same frequency

Cf. A045540, A052047, A052048, A052049, A052050, A052051, A052052.

Cf. A078255.

filter:= proc(n) local L,M;
  L:= convert(n,base,10);
  M:= {seq(numboccur(i,L),i=0..9)} minus {0};
  nops(M) = 1
end proc:
select(filter, [seq(i^2,i=0..200)]); # Robert Israel, Jan 08 2018

t={}; Do[If[Length[DeleteDuplicates[Transpose[Tally[IntegerDigits[n^2]]][[2]]]] == 1, AppendTo[t,n^2]], {n,0,80}]; t (* Jayanta Basu, May 10 2013 *)
sfQ[n_]:=Length[Union[Select[DigitCount[n],#!=0&]]]==1; Select[ Range[ 0,80]^2,sfQ] (* Harvey P. Dale, May 05 2019 *)

Offset corrected by Michel Marcus, Aug 12 2015

A052049 a(n)^2 is a square whose digits occur with an equal minimum frequency of 2.

Original entry on oeis.org

88, 478, 577, 583, 715, 836, 880, 881, 893, 3362, 3386, 3911, 4077, 4780, 5077, 5239, 5369, 5770, 5784, 5789, 5830, 5858, 6523, 6756, 6772, 6926, 6941, 7107, 7150, 7359, 7535, 7827, 8043, 8196, 8229, 8360, 8525, 8810, 8930, 8989, 9251, 9701, 9764, 9786

Offset: 1

Patrick De Geest, Dec 15 1999

There are A225428(10) = 597959 terms in this sequence. The last term is 9994363488, whose square is 99887301530267526144 = A052050(597959). - Hugo Pfoertner, May 12 2023

			577^2 = 332929, which contains each of its digits (2, 3, and 9) twice, so 577 is in this sequence.

Nathaniel Johnston, Table of n, a(n) for n = 1..5000
Patrick De Geest, Numbers whose digits occur with same frequency

Cf. A045540, A052046, A052047, A052048, A052050, A052051, A052052, A225428.

isA052049 := proc(n) local d, k, fr, eqfr: d:=convert(n^2, base, 10): eqfr:=true: fr:=numboccur(d[1], d): if(fr=1)then return false: fi: for k from 0 to 9 do if(not member(numboccur(k, d), {fr, 0}))then eqfr:=false: break: fi: od: return eqfr: end: seq(`if`(isA052049(n), n, NULL), n=1..9800); # Nathaniel Johnston, Jun 02 2011

ta[n_]:=DeleteDuplicates[Transpose[Tally[IntegerDigits[n^2]]][[2]]]; t ={}; Do[If[Length[x=ta[n]]==1 && x[[1]]>=2, AppendTo[t,n]],{n,9800}]; t (* Jayanta Basu, May 11 2013 *)

A052050 Squares whose digits occur with an equal minimal frequency of 2.

Original entry on oeis.org

7744, 228484, 332929, 339889, 511225, 698896, 774400, 776161, 797449, 11303044, 11464996, 15295921, 16621929, 22848400, 25775929, 27447121, 28826161, 33292900, 33454656, 33512521, 33988900, 34316164, 42549529, 45643536, 45859984, 47969476, 48177481, 50509449, 51122500

Offset: 1

Patrick De Geest, Dec 15 1999

There are A225428(10)=597959 terms in this sequence. The last term is 99887301530267526144 = 9994363488^2. - Hugo Pfoertner, May 12 2023

Patrick De Geest, Numbers whose digits occur with same frequency

Cf. A052049, A045540, A052046, A052047, A052048, A052051, A052052, A225428.

Select[Table[n^2, {n, 6760}], Union[Last[Transpose[Tally[IntegerDigits[#]]]]] == {2} &] (* Jayanta Basu, Jun 17 2013 *)

from itertools import islice
from collections import Counter
def afull(): yield from (x**2 for x in range(10**10) if set(Counter(str(x**2)).values()) == {2})
print(list(islice(afull(), 47))) # Michael S. Branicky, May 12 2023

A052052 Cubes whose digits occur with an equal minimal frequency of 2.

Original entry on oeis.org

1331, 238328, 27818127, 700227072, 2815166528, 4861163384, 8972978552, 107510668875, 157376536199, 236143627741, 246963938824, 267463420307, 748118742552, 834034807997, 915215787829, 10809986553631, 11882007579259

Offset: 1

Patrick De Geest, Dec 15 1999

			888^3 = 700227072 and digits 0,2 and 7 each occur thrice.

P. De Geest, Numbers whose digits occur with same frequency

Cf. A052051, A045540, A052046, A052047, A052048, A052049, A052050.

Offset corrected by Michel Marcus, Aug 12 2015

A052047 Numbers k such that the digits of k^3 occur with the same frequency.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 13, 16, 17, 18, 19, 21, 22, 24, 27, 29, 32, 35, 38, 41, 59, 62, 66, 69, 73, 75, 76, 84, 88, 93, 97, 135, 145, 203, 289, 297, 302, 303, 319, 888, 1412, 1694, 2078, 4755, 5399, 6181, 6274, 6443, 9078, 9413, 9709, 22111, 22819

Offset: 1

Patrick De Geest, Dec 15 1999

Robert Israel, Table of n, a(n) for n = 1..500
P. De Geest, Numbers whose digits occur with same frequency

Cf. A052048, A045540, A052046, A052049, A052050, A052051, A052052.

filter:= proc(n) local L,V,k;
  L:= convert(n^3,base,10);
  V:= {seq(numboccur(k,L),k=0..9)};
  nops(V minus {0}) = 1
end proc:filter(0):= true:select(filter, [$0..10000]); # Robert Israel, Mar 12 2020

t={}; Do[If[Length[DeleteDuplicates[Transpose[Tally[IntegerDigits[n^3]]][[2]]]]==1,AppendTo[t,n]],{n,0,23000}]; t (* Jayanta Basu, May 11 2013 *)
Select[Range[0,23000],Length[Union[DeleteCases[DigitCount[#^3],0]]]==1&] (* Harvey P. Dale, Jun 14 2020 *)

A052048 Cubes whose digits occur with the same frequency.

Original entry on oeis.org

0, 1, 8, 27, 64, 125, 216, 512, 729, 1331, 1728, 2197, 4096, 4913, 5832, 6859, 9261, 10648, 13824, 19683, 24389, 32768, 42875, 54872, 68921, 205379, 238328, 287496, 328509, 389017, 421875, 438976, 592704, 681472, 804357, 912673, 2460375, 3048625

Offset: 1

Patrick De Geest, Dec 15 1999

			E.g., 238328 = 62^3 -> digits 2, 3 and 8 occur each twice.

Nathaniel Johnston, Table of n, a(n) for n = 1..450
Patrick De Geest, Numbers whose digits occur with same frequency

Cf. A052047, A045540, A052046, A052049, A052050, A052051, A052052.

isA052048 := proc(n) local d,k,fr,eqfr: if(n=0)then return true: fi: d:=convert(n^3,base,10): eqfr:=true: fr:=numboccur(d[1],d): for k from 0 to 9 do if(not member(numboccur(k,d),{fr,0}))then eqfr:=false: break: fi: od: return eqfr: end: seq(`if`(isA052048(n), n^3, NULL), n=0..200); # Nathaniel Johnston, May 31 2011

t={}; Do[If[Length[DeleteDuplicates[Transpose[Tally[IntegerDigits[n^3]]][[2]]]]==1,AppendTo[t,n^3]],{n,0,200}]; t (* Jayanta Basu, May 11 2013 *)

a(n) = A052047(n)^3. - Andrew Howroyd, Aug 11 2024

Offset corrected by Andrew Howroyd, Aug 11 2024

A052051 Numbers k such that k^3 is a cube whose digits occur with an equal minimum frequency of 2.

Original entry on oeis.org

11, 62, 303, 888, 1412, 1694, 2078, 4755, 5399, 6181, 6274, 6443, 9078, 9413, 9709, 22111, 22819, 23894, 24835, 26636, 26881, 28895, 29631, 30593, 32069, 32687, 32723, 32887, 34699, 35042, 36536, 36705, 36869, 37568, 40675, 41538, 41674

Offset: 1

Patrick De Geest, Dec 15 1999

P. De Geest, Numbers whose digits occur with same frequency

Cf. A052052, A045540, A052046, A052047, A052048, A052049, A052050.

A119509 Positive numbers whose square contains no digit more than once.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 16, 17, 18, 19, 23, 24, 25, 27, 28, 29, 31, 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, 113, 116, 117, 118, 124, 126, 128, 133

Offset: 1

Tanya Khovanova, Jul 26 2006

There are exactly 610 terms. a(610) = 99066 and 99066^2 = 9814072356. - Rick L. Shepherd, Jul 27 2006

If we count 0, there is one more term, for a total of 611. - T. D. Noe, Jun 21 2013

Rick L. Shepherd, Table of n, a(n) for n = 1..610 (full sequence)

Subsequence of A045540 = numbers whose squares contain an equal number of each digit that they contain. The first number that belongs to A045540 and doesn't belong to this sequence is number 88.

Cf. A078255, A036745, A075309, A162950.

[n: n in [1..10^5] | #Set(d) eq #d where d is Intseq(n^2)];  // Bruno Berselli, Aug 02 2011

lim:=floor(sqrt(9876543210)): A119509:={}: for n from 1 to lim do pandig:=true: d:=convert(n^2,base,10): for k from 0 to 9 do if(numboccur(k, d)>1)then pandig:=false: break: fi: od: if(pandig)then A119509 := A119509 union {n}: fi: od: op(sort(convert(A119509,list))); # Nathaniel Johnston, Jun 23 2011

Select[Range[1000000], Length[IntegerDigits[ # ^2]] == Length[Union[IntegerDigits[ # ^2]]] &] (* Tanya Khovanova, May 29 2007 *)
Select[Range[10^5], Max[DigitCount[#^2]] <= 1 &] (* T. D. Noe, Aug 02 2011 *)

is_A119509(n)=#(n=digits(n^2))==#Set(n) \\ M. F. Hasler, Sep 08 2017

def ok(n): s = str(n**2); return n > 0 and len(set(s)) == len(s)
afull = [k for k in range(10**5) if ok(k)] # Michael S. Branicky, Nov 27 2022

More terms from Rick L. Shepherd, Jul 27 2006

A052069 a(n)^2 is the smallest square whose digits occur with the same frequency n.

Original entry on oeis.org

0, 88, 10011, 31646191, 16431563, 667567716, 10715008859, 652246443112, 15647628653832, 781035313645040

Offset: 1

Patrick De Geest, Jan 15 2000

			31646191^2 = 1001481404808481 and its digits 0, 1, 4 and 8 each occur four times.

Patrick De Geest, Numbers whose digits occur with same frequency

Cf. A052070, A045540, A052046, A052049, A052050, A052071.

Table[i = 0;
While[x = i^2; Union@DeleteCases[DigitCount[x], 0] != {n}, i++];
i, {n, 10}] (* Robert Price, Oct 12 2019 *)

2 more terms from Jon E. Schoenfield, Aug 18 2007

a(10) from Giovanni Resta, Aug 19 2018

A052070 Smallest square all of whose digits occur with the same frequency n.

Original entry on oeis.org

0, 7744, 100220121, 1001481404808481, 269996262622969, 445646655445456656, 114811414848448481881, 425425422552255452244544, 244848282488224248488284224, 610016161160606006011116601600

Offset: 1

Patrick De Geest, Jan 15 2000

			1001481404808481 (= 31646191^2) and its digits 0, 1, 4 and 8 each occur four times.

Patrick De Geest, Numbers whose digits occur with same frequency

Cf. A052069, A045540, A052046, A052049, A052050, A052072.

Table[i = 0;
While[x = i^2; Union@DeleteCases[DigitCount[x], 0] != {n}, i++];
x, {n, 10}] (* Robert Price, Oct 12 2019 *)

Two more terms from Jon E. Schoenfield, Oct 11 2008
Offset corrected by Michel Marcus, Aug 12 2015
a(10) from Giovanni Resta, Aug 19 2018

cp's OEIS Frontend

A052046 Squares whose digits occur with the same frequency.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A052049 a(n)^2 is a square whose digits occur with an equal minimum frequency of 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A052050 Squares whose digits occur with an equal minimal frequency of 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

A052052 Cubes whose digits occur with an equal minimal frequency of 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A052047 Numbers k such that the digits of k^3 occur with the same frequency.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A052048 Cubes whose digits occur with the same frequency.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A052051 Numbers k such that k^3 is a cube whose digits occur with an equal minimum frequency of 2.

Views

Author

Keywords

Links

Crossrefs

A119509 Positive numbers whose square contains no digit more than once.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI