A052046 -id:A052046 - cp's OEIS frontend

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

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

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.

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

A378492 Squares where larger digits have larger multiplicity.

Original entry on oeis.org

0, 1, 4, 9, 144, 441, 1444, 29929, 55225, 166464, 255025, 299209, 633616, 646416, 767376, 4999696, 9696996, 34433424, 228281881, 414041104, 414488881, 424442404, 536663556, 969699600, 1649496996, 1929229929, 2636206336, 2666999449, 2929299129, 2996029696, 4664343616

Offset: 1

Erich Friedman, Nov 28 2024

Alois P. Heinz, Table of n, a(n) for n = 1..500

Cf. A000290, A018884, A052046, A235718, A378498.

filter:= proc(n) local L,S;
   L:= convert(n,base,10);
   S:= Statistics:-Tally(L,output=list);
   S:= sort(S, (a,b) -> lhs(a) < lhs(b));
   andmap(t -> rhs(S[t])Robert Israel, Nov 29 2024

increasingQ[L_]:=Min[Rest[(L-RotateRight[L])]]>0;
sortedQ[n_]:=increasingQ[Sort[Tally[IntegerDigits[n]]][[All,2]]]
Select[Range[575000000]^2,sortedQ]

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

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

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

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