A030098 -id:A030098 - cp's OEIS frontend

A030097 Numbers k such that k^2 has only even digits.

0, 2, 8, 20, 22, 68, 78, 80, 92, 162, 168, 200, 202, 220, 262, 298, 478, 492, 498, 668, 680, 780, 800, 898, 908, 920, 932, 1422, 1492, 1498, 1620, 1680, 1692, 2000, 2002, 2020, 2022, 2192, 2200, 2202, 2498, 2502, 2578, 2620, 2832, 2878, 2978, 2980, 4502

Offset: 1

Patrick De Geest

A136904 is a subsequence. - Zak Seidov, Dec 03 2012

Zak Seidov, Table of n, a(n) for n = 1..2000

Cf. A030098, A136904.

Select[ Range[ 3000 ], Union[ EvenQ[ IntegerDigits[ #^2 ] ] ] == {True} & ]

is_A030097(n)=!setminus(Set(Vec(Str(n^2))),Vec("02468"))  \\ M. F. Hasler, Dec 13 2011

a(n) = sqrt(A030098(n)). - Zak Seidov, Dec 03 2012

More terms from Zak Seidov, May 24 2010

A030288 a(n+1) is smallest square > a(n) having no digits in common with a(n), with a(0) = 0.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 81, 225, 361, 400, 529, 676, 841, 900, 1156, 2209, 3136, 4225, 6889, 7225, 8100, 24336, 58081, 69696, 70225, 84681, 90000, 111556, 200704, 316969, 407044, 511225, 608400, 923521, 4000000, 5112121, 6036849

Offset: 0

Patrick De Geest

It appears that from a(102) on, there is a 4-periodic pattern: a(4k) ~ 3*10^(k-3) a(4k+1) ~ 6.1111...*10^(k-3), a(4k+2) ~ 7*10^(k-3), a(4k+3) ~ 8.1111...*10^(k-3), where ~ means the next larger square which has only digits {0, 3, 4, 5, 7} for even-indexed terms, or {1, 2, 6, 8, 9} for odd-indexed terms. - M. F. Hasler, Nov 12 2017

David W. Wilson and Jon E. Schoenfield, Table of n, a(n) for n = 0..250 (first 231 terms from David W. Wilson)

Cf. A000290, A019544, A030098, A030287, A030289.

FromDigits /@ NestList[Block[{k = Sqrt@ FromDigits@ # + 1, m}, While[ContainsAny[#, Set[m, IntegerDigits[k^2]]], k++]; m] &, {0}, 38] (* Michael De Vlieger, Nov 02 2017 *)
ssga[a_]:=Module[{k=Floor[Sqrt[a]]+1},While[Length[Intersection[IntegerDigits[k^2],IntegerDigits[ a]]]> 0,k++];k^2]; NestList[ssga,0,40] (* Harvey P. Dale, Sep 10 2024 *)

next_A030288(n, D(n)=Set(digits(n)), S=D(n))={for(k=sqrtint(n)+1, oo, #setintersect(D(k^2), S)||return(k^2))} \\ Could be made more efficient by implementing the observed patterns, in particular for n >= 104. - M. F. Hasler, Nov 12 2017

a(n) = A030287(n)^2. - Michel Marcus, Nov 03 2017

A343724 a(n) is the smallest n-digit square with all digits even.

Original entry on oeis.org

0, 64, 400, 4624, 26244, 228484, 2022084, 20268004, 202208400, 2006860804, 20220840000, 200084446864, 2002004266084, 20000286620224, 200080402620484, 2000028662022400, 20000086482842244, 200002866202240000, 2000008648284224400, 20000246442286866064

Offset: 1

Jon E. Schoenfield, May 19 2021

From Robert G. Wilson v, May 20 2021: (Start)
The square root of any term is == {0, 2, 8} (mod 10).
Other than 1 and 9, there are no squares which contain only odd digits.
(End)

Chai Wah Wu, Table of n, a(n) for n = 1..61 (terms for n = 1..40 from Robert G. Wilson v)

Cf. A030098, A343725.

a[n_] := Block[{k = Floor[ Sqrt[10^n/5]]}, If[OddQ@k, k--]; While[ Union[ EvenQ[ IntegerDigits[ k^2]]] != {True}, k += 2]; k^2]; Array[ a, 20] (* Robert G. Wilson v, May 20 2021 *)

A343725 a(n) is the largest n-digit square with all digits even.

Original entry on oeis.org

4, 64, 484, 8464, 88804, 868624, 8880400, 86862400, 888040000, 8880800644, 88864802404, 888868068804, 8886826042084, 88886806880400, 888686062262244, 8888680688040000, 88868606226224400, 888868068804000000, 8888600480200862404, 88886806880400000000

Offset: 1

Jon E. Schoenfield, May 19 2021

Chai Wah Wu, Table of n, a(n) for n = 1..63

Cf. A030098, A343724.

A103751 Squares whose digits are all positive and even.

Original entry on oeis.org

4, 64, 484, 4624, 8464, 26244, 28224, 68644, 228484, 446224, 824464, 868624, 2862864, 8282884, 8868484, 22448644, 26646244, 44462224, 82228624, 82664464, 222248464, 284866884, 662444644, 866242624, 4246868224, 4444622224, 6266622244, 6282464644, 6668682244, 8264264464, 8268628624

Offset: 1

Emeric Deutsch, Mar 28 2005

Subset of A030098.
All terms end with 4, because when k^2 ends with 6, the tens digit of k^2 is always odd. - Bernard Schott, May 02 2022
The sequence is infinite because squares of the form 4 = 2^2, 64 = 8^2, 4624 = 68^2, 446224 = 668^2, 44462224 = 6668^2, ... (2*(10^k + 2) / 3 )^2 , k >= 0, are terms. - Marius A. Burtea, May 02 2022

Michael S. Branicky, Table of n, a(n) for n = 1..4198 (terms 1..180 from Marius A. Burtea)

Cf. A030098.

[n:n in [s*s:s in [1..100000]]| Set(Intseq(n)) subset {2,4,6,8}]; // Marius A. Burtea, May 02 2022

a:=proc(n) if convert(convert((n^2),base,10),set) subset {2,4,6,8} then n^2 else fi end:seq(a(n),n=1..100000);

pevQ[n_]:=Module[{idn=IntegerDigits[n]},FreeQ[idn,0]&&And@@EvenQ[idn]]; Select[Range[70000]^2,pevQ] (* Harvey P. Dale, Jul 19 2013 *)

isok(n) = my(d=digits(n)); vecmin(d) && (#select(x->(x%2), d) == 0);
lista(nn) = {my(list = List()); for (n=1, nn, if (isok(n^2), listput(list, n^2););); Vec(list);} \\ Michel Marcus, May 02 2022

More terms from Bernard Schott, May 02 2022

A343726 Squares with exactly one even digit.

Original entry on oeis.org

0, 4, 16, 25, 36, 49, 81, 121, 169, 196, 361, 529, 576, 729, 961, 1156, 1369, 1521, 1936, 3136, 3721, 3969, 5329, 5776, 5929, 7396, 7569, 7921, 15129, 15376, 17161, 17956, 19321, 31329, 35721, 51529, 53361, 57121, 59536, 97969, 111556, 113569, 119716, 131769

Offset: 1

Jon E. Schoenfield, May 19 2021

The even digit is always one of the last two digits.

The only squares with no digits even are the one-digit odd squares 1 and 9.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A000290, A030098, A118070, A343724, A343725.

q:= n-> (l-> is(add(i mod 2, i=l)=nops(l)-1))(convert(n, base, 10)):
select(q, [i^2$i=0..400])[];  # Alois P. Heinz, May 22 2021

Select[Range[0, 400]^2, Count[IntegerDigits[#], ?EvenQ] == 1 &] (* _Amiram Eldar, May 21 2021 *)

isA343726(n) = if(issquare(n) && (n!=0), my(d=digits(n)); #d - vecsum(d%2) == 1, n==0) \\ Jianing Song, May 22 2021

def ok(sq): return sum(d in "02468" for d in str(sq)) == 1
def aupto(limit):
  sqs = (i*i for i in range(int(limit**.5)+2) if i*i <= limit)
  return list(filter(ok, sqs))
print(aupto(131769)) # Michael S. Branicky, May 20 2021

Intersection of A000290 and A118070.

A343728 Numbers with all digits even whose squares have all but one digit odd.

Original entry on oeis.org

0, 2, 4, 6, 24, 44, 86, 244, 424, 444, 846, 2444, 4424, 6286, 42424, 44244, 240244, 244086, 244866, 268286, 420846, 442244, 446286, 628646, 880646, 2402444, 4402044, 4442244, 8448666, 24040244, 24064866, 26682086, 26682866, 26828666, 28244244, 42400424

Offset: 1

Jon E. Schoenfield, May 20 2021

Of course, the one even digit in the square is always the last digit.

			244086 is a term: all its digits are even, and 244086^2 = 59577975396 has all but one digit odd.
244044086 is a term: all its digits are even, and 244044086^2 = 59557515911575396 has all but one digit odd.

Chai Wah Wu, Table of n, a(n) for n = 1..100

Cf. A014263, A030098, A343727.

Select[Range[0, 10^6], AllTrue[IntegerDigits[#], EvenQ] && AllTrue[Most @ IntegerDigits[#^2], OddQ] &] (* Amiram Eldar, May 20 2021 *)

def ok(n):
  r, s = str(n), str(n*n)
  return all(d in "02468" for d in r) and all(d in "13579" for d in s[:-1])
print(list(filter(ok, range(0, 42400425, 2)))) # Michael S. Branicky, May 20 2021

from gmpy2 import digits
A343728_list = [n for n in (2*int(digits(d,5)) for d in range(10**6)) if set(str(n**2)[:-1]) <= set('13579')] # Chai Wah Wu, May 21 2021

A343727 Numbers with all digits odd whose squares have only one odd digit.

Original entry on oeis.org

1, 3, 5, 7, 9, 15, 17, 51, 53, 79, 91, 93, 151, 155, 157, 533, 535, 775, 779, 793, 917, 1557, 1571, 1575, 5179, 5333, 5335, 7759, 7799, 9317, 9393, 9395, 15557, 15559, 15755, 51595, 53179, 53333, 53335, 77759, 79151, 79175, 93917, 151151, 155135, 155191

Offset: 1

Jon E. Schoenfield, May 20 2021

Of course, the one odd digit in the square is always the last digit.

The sequence is infinite because it contains the family of numbers 5, 53, 533, 5333, ....... with squares 25, 2809, 284089, 28440889, 2844408889. .... and respectively 535, 5335, 53335, ... with squares 286225, 28462225, 2844622225, 284446222225, ... - Marius A. Burtea, May 21 2021

			53179 is a term: all its digits are odd, and 53179^2 = 2828006041 has only one odd digit.
15113133375599 is a term: all its digits are odd, and 15113133375599^2 = 228406800428644424408608801 has only one odd digit.

Cf. A014261, A030098, A343726, A343728.

[n:n in [1..160000 by 2]|Set(Intseq(n)) subset {1,3,5,7,9} and Set(Intseq(n*n div 10)) subset {0,2,4,6,8}]; // Marius A. Burtea, May 21 2021

Select[Range[160000], AllTrue[IntegerDigits[#], OddQ] && AllTrue[Most @ IntegerDigits[#^2], EvenQ] &] (* Amiram Eldar, May 20 2021 *)

def ok(n):
  r, s = str(n), str(n*n)
  return all(d in "13579" for d in r) and all(d in "02468" for d in s[:-1])
print(list(filter(ok, range(1, 155192, 2)))) # Michael S. Branicky, May 20 2021

from itertools import product
A343727_list = [n for n in (int(''.join(d)) for l in range(1,6) for d in product('13579',repeat=l)) if set(str(n**2)[:-1]) <= set('02468')] # Chai Wah Wu, May 21 2021

A349460 Squares composed of digits {0,2,4}.

Original entry on oeis.org

0, 4, 400, 40000, 2244004, 4000000, 42224004, 224400400, 400000000, 424442404, 4222400400, 22200404004, 22440040000, 40000000000, 42444240400, 422240040000, 2220040400400, 2244004000000, 4000000000000, 4024044024004, 4244424040000, 40244204444224, 42224004000000

Offset: 1

Daniel Blam, Nov 18 2021

From Marius A. Burtea, Nov 18 2021: (Start)
The sequence is infinite because if m > 0 is a term, then 100*m is also a term.
Also, the squares of the numbers 20602, 2006002, 200060002, ..., (2*10^(2*k) + 6*10^k + 2), k >= 2, are 424442404, 4024044024004, 40024004400240004, 400024000440002400004, ... and have only the digits 0, 2 and 4 and are not divisible by 100. (End)

Subsequence of A000290 and A030098.

[n : n in [s*s:s in [1..1500000]]|Set(Intseq(n)) subset {0,2,4}]; // Marius A. Burtea, Nov 18 2021

Select[Range[0, 10^7, 2]^2, AllTrue[IntegerDigits[#], MemberQ[{0, 2, 4}, #1] &] &] (* Amiram Eldar, Nov 18 2021 *)

from itertools import islice, count
def A349460(): return filter(lambda n: set(str(n)) <= {'0','2','4'},(n*n for n in count(0)))
A349460_list = list(islice(A349460(),20)) # Chai Wah Wu, Nov 19 2021

A202172 Squares containing only the digits 0, 4 or 8.

Original entry on oeis.org

0, 4, 400, 484, 40000, 40804, 48400, 88804, 4000000, 4008004, 4080400, 4088484, 4840000, 4848804, 8880400, 400000000, 400080004, 400800400, 400880484, 408040000, 408848400, 484000000, 484088004, 484880400, 840884004, 888040000, 40000000000, 40000800004

Offset: 1

M. F. Hasler, Dec 13 2011

Cf. A030098.

Select[Range[0, 400000]^2, Complement[Union[IntegerDigits[#]], {0, 4, 8}] == {} &] (* T. D. Noe, Dec 21 2011 *)
Select[FromDigits/@Tuples[{0,4,8},11],IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, Apr 19 2019 *)

for(i=0,99999,setminus(Set(Vec(Str(i^2))),Vec("048")) || print1(i^2,","))

a(n)=A202170(n)^2

cp's OEIS Frontend

A030097 Numbers k such that k^2 has only even digits.

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A030288 a(n+1) is smallest square > a(n) having no digits in common with a(n), with a(0) = 0.

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A343724 a(n) is the smallest n-digit square with all digits even.

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A343725 a(n) is the largest n-digit square with all digits even.

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A103751 Squares whose digits are all positive and even.

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A343726 Squares with exactly one even digit.

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A343728 Numbers with all digits even whose squares have all but one digit odd.

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A343727 Numbers with all digits odd whose squares have only one odd digit.

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