A052051 -id:A052051 - 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 *)

A045540 Numbers whose square contains an equal number of each digit that it contains.

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, 88, 89, 93, 95, 96, 98, 99, 113, 116, 117, 118, 124, 126, 128, 133, 134, 136

Offset: 1

Erich Friedman

The sequence is expected to be infinite. Heuristically, if m is divisible by 10 there should be approximately constant * 10^(m/2)/m^(9/2) m-digit squares where all 10 digits have frequency m/10. - Robert Israel, Aug 14 2015

Robert Israel, Table of n, a(n) for n = 1..3000
P. De Geest, Numbers whose digits occur with same frequency

Cf. A052046, A052047, A052048, A052049, A052050, A052051, A052052, A052060.

filter:= proc(n) local x,i,P;
P:= add(x^i, i=convert(n^2,base,10));
nops({coeffs(P,x)}) = 1
end proc:
select(filter, [$1..10^4]); # Robert Israel, Aug 14 2015

t={}; Do[If[Length[DeleteDuplicates[Transpose[Tally[IntegerDigits[n^2]]][[2]]]]==1,AppendTo[t,n]],{n,136}]; t (* Jayanta Basu, May 10 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

A052071 a(n)^3 is the smallest cube whose digits occur with the same frequency n.

Original entry on oeis.org

0, 11, 888, 2830479, 120023142, 6351783105, 267745815817, 13280759167047

Offset: 1

Patrick De Geest, Jan 15 2000

			2830479^3 = 22676697737363992239 and its digits 2, 3, 6, 7 and 9 each occur four times.

Patrick De Geest, Numbers whose digits occur with same frequency

Cf. A052072, A052047, A052048, A052051, A052052, A052069.

Table[i = 0;
While[x = i^3; Union@DeleteCases[DigitCount[x], 0] != {n}, i++];
i, {n, 7}] (* Robert Price, Oct 12 2019 *)

Offset corrected by Michel Marcus, Aug 12 2015
a(7) from Giovanni Resta, Aug 19 2018
a(8) from Mihael Tunik, May 26 2025

A052072 Smallest cubes whose digits occur with the same frequency n.

Original entry on oeis.org

0, 1331, 700227072, 22676697737363992239, 1728999927211172788179288, 256263633328535368685258882625, 19194114355415391344355399945943513, 2342441231418833141828248411332348812823

Offset: 1

Patrick De Geest, Jan 15 2000

			22676697737363992239 (= 2830479^3) and its digits 2, 3, 6, 7 and 9 each occur four times.

Patrick De Geest, Numbers whose digits occur with same frequency

Cf. A052071, A052047, A052048, A052051, A052052, A052070.

Table[i = 0;
While[x = i^3; Union@DeleteCases[DigitCount[x], 0] != {n}, i++];
x, {n, 7}] (* Robert Price, Oct 12 2019 *)
Select[Range[400]^3,Length[Union[DeleteCases[DigitCount[#],0]]]==1&] (* Harvey P. Dale, Apr 30 2022 *)

def A052072(n):
    a, b, c = 0, 0, 0
    for i in range(10**9):
        s = str(c)
        for d in set(s):
            if s.count(d) != n:
                break
        else:
            return c
        c += a + b + 1
        b += 2*a + 3
        a += 3
    return 'search limit reached.'  # Chai Wah Wu, Aug 11 2015

Offset corrected by Michel Marcus, Aug 12 2015
a(7) from Giovanni Resta, Aug 19 2018
a(8) from Mihael Tunik, May 26 2025

A308708 Numbers k such that k^3 contains exactly three distinct digits; numbers with trailing zeros are excluded.

Original entry on oeis.org

5, 6, 8, 9, 14, 15, 36, 62, 92, 101, 173, 192, 211, 888, 1001, 3543, 10001, 100001, 110011, 146796, 1000001, 10000001, 100000001, 1000000001, 10000000001

Offset: 1

Andrej Jakobcic, Aug 01 2019

10^k + 1 (A000533(k)) is a term for k >= 2. - Jinyuan Wang, Aug 02 2019

			a(8) = 62 because 62^3 = 238328, which contains exactly three distinct digits.

Cf. A000533, A202940, A030294, A052051.

[k:k in [1..10000001]| k mod 10 ne 0 and  #Set(Intseq(k^3)) eq 3]; // Marius A. Burtea, Aug 02 2019

Select[Range[10001], Mod[#, 10] > 0 && Length@ Union@ IntegerDigits[#^3] == 3 &] (* Giovanni Resta, Sep 05 2019 *)

is(k) = #vecsort(digits(k^3), , 8)==3 && k%10!=0; \\ Jinyuan Wang, Aug 02 2019

More terms from Jinyuan Wang, Aug 02 2019

a(23)-a(25) from Jon E. Schoenfield, Aug 02 2019

cp's OEIS Frontend

A052046 Squares whose digits occur with the same frequency.

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

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A045540 Numbers whose square contains an equal number of each digit that it contains.

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

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

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

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A052048 Cubes whose digits occur with the same frequency.

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A052071 a(n)^3 is the smallest cube whose digits occur with the same frequency n.

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