A030097 -id:A030097 - cp's OEIS frontend

A030098 Squares whose digits are all even.

0, 4, 64, 400, 484, 4624, 6084, 6400, 8464, 26244, 28224, 40000, 40804, 48400, 68644, 88804, 228484, 242064, 248004, 446224, 462400, 608400, 640000, 806404, 824464, 846400, 868624, 2022084, 2226064, 2244004, 2624400, 2822400, 2862864, 4000000, 4008004, 4080400

Offset: 1

Patrick De Geest

On the other hand, the only squares whose digits are all odd are 1 and 9, because the tens digit of all odd squares >= 25 (A016754) is always even. - Bernard Schott, Jan 24 2023

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Subsequence of A075787.

Cf. A016754, A030097.

t = {}; n = -1; While[Length[t] < 1000, n++; If[Intersection[IntegerDigits[n^2], {1, 3, 5, 7, 9}] == {}, AppendTo[t, n^2]]] (* T. D. Noe, Apr 03 2014 *)
Select[Range[0,3000]^2,AllTrue[IntegerDigits[#],EvenQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 19 2016 *)

from math import isqrt
def ok(sq): return all(d in "02468" for d in str(sq))
def aupto(limit):
  sqs = (i*i for i in range(0, isqrt(limit)+1, 2))
  return list(filter(ok, sqs))
print(aupto(4080400)) # Michael S. Branicky, May 20 2021

a(n) = A030097(n)^2. - Michel Marcus, Apr 03 2014

A202170 Numbers k such that k^2 has only digits 0, 4 and 8.

Original entry on oeis.org

0, 2, 20, 22, 200, 202, 220, 298, 2000, 2002, 2020, 2022, 2200, 2202, 2980, 20000, 20002, 20020, 20022, 20200, 20220, 22000, 22002, 22020, 28998, 29800, 200000, 200002, 200020, 200022, 200200, 200202, 200220, 202000, 202002, 202200, 220000, 220002, 220020, 220200, 289980, 298000, 632522, 2000000

Offset: 1

M. F. Hasler, Dec 13 2011

k = 2*m where m^2 has only digits 0, 1 and 2. - Robert Israel, Jul 17 2018

Robert Israel, Table of n, a(n) for n = 1..204

Cf. A030097, A058441.

Includes 2*A136808.

[ n: n in [0..10^6 by 2] | Set(Intseq(n^2)) subset [0,4,8] ]; // Bruno Berselli, Dec 19 2011

Res:= NULL:
for x from 0 to 3^12 do
  L:= convert(x,base,3);
  y:= add(L[i]*10^(i-1),i=1..nops(L));
  if issqr(y) then
    Res:= Res, 2*sqrt(y)
  fi
od:
Res; # Robert Israel, Jul 17 2018

Select[Sqrt[FromDigits[#]]&/@Tuples[{0,4,8},13],IntegerQ] (* Harvey P. Dale, Sep 08 2024 *)

is_A202170(n)=!setminus(Set(Vec(Str(n^2))),Vec("048"))

More terms from Robert Israel, Jul 17 2018

A385300 Integers k containing only odd digits, except the last digit, such that k^2 is composed of only even digits.

Original entry on oeis.org

2, 8, 78, 92, 932, 7798, 51192, 939398, 5157798, 7797578, 7797978, 9393978, 15119592, 15773398, 53179378, 53559332, 77979998, 79175932, 155139378, 157759592, 517751738, 535393932, 917933998, 939597798, 1511979592, 5157759592, 7771757978, 7775735378, 9393959798

Offset: 1

Gonzalo Martínez, Jun 24 2025

Although the terms in this list contain only odd digits, except for the units digit, their squares have only even digits.
The terms of the list are even numbers ending only in 2 or in 8. Proof: if n ends in 0, 4 or 6, since the penultimate digit of n is odd, then the last 2 digits of n can be: 10, 14, 16, 30, 34, 36, 50, 54, 56, 70, 74, 76, 90, 94 and 96. If n ends in 10, 30, 50, 70 or 90, then the antepenultimate digit of n^2 is odd, while in the other cases the penultimate digit of n^2 is odd.
a(n) == 32, 38, 78, 92, 98 (mod 100), for n > 3.
a(n) == 192, 332, 338, 378, 398, 578, 592, 738, 798, 932, 978, 992, 998 (mod 1000), for n > 4. - Chai Wah Wu, Jun 25 2025

			7798 is a term, since only the last digit is even and 7798^2 = 60808804, whose digits are all even.

David A. Corneth, Table of n, a(n) for n = 1..360
David A. Corneth, PARI program

Cf. A000290, A014261, A030097, A136904.

Select[Range[10^7],NoneTrue[Take[IntegerDigits[#],IntegerLength[#]-1],EvenQ]&&AllTrue[IntegerDigits[#^2],EvenQ]&] (* James C. McMahon, Jun 28 2025 *)

odd(n) = {my(k=1); while(n>5^k, n-=5^k; k++); fromdigits([2*d+1 | d<-digits(5^k+n-1, 5)]) - 3*10^k}; \\ A014261
lista(nn) = my(list=List()); forstep (e=2, 8, 2, if (#select(x->(x%2), digits(e^2)) == 0, listput(list, e));); for (n=1, nn, my(o=odd(n)); forstep (e=0, 8, 2, my(oe = 10*o+e); if (#select(x->(x%2), digits(oe^2)) == 0, listput(list, oe)););); Vec(list); \\ Michel Marcus, Jun 25 2025

from itertools import count, product, islice
def A385300_gen(): # generator of terms
    for l in count(1):
        for d in product('13579',repeat=l):
            m = int(''.join(d))-1
            if m>0 and set(str(m**2))<={'0','2','4','6','8'}:
                yield m
A385300_list = list(islice(A385300_gen(),29)) # Chai Wah Wu, Jun 25 2025

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A030098 Squares whose digits are all even.

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A202170 Numbers k such that k^2 has only digits 0, 4 and 8.

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A385300 Integers k containing only odd digits, except the last digit, such that k^2 is composed of only even digits.

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Author

Keywords

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Crossrefs

Programs

Mathematica

PARI

Python