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

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

Author

Gonzalo Martínez, Jun 24 2025

Keywords

• nonn
• base

Comments

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

Examples

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

Links

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

Crossrefs

Cf. A000290, A014261, A030097, A136904.

Programs

Mathematica

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

PARI

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

Python

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