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A267765
Numbers whose base-5 representation is a square when read in base 10.
2
0, 1, 4, 25, 36, 49, 89, 100, 121, 139, 249, 329, 351, 625, 676, 729, 900, 961, 1225, 1551, 1654, 2146, 2225, 2289, 2500, 2601, 3025, 3289, 3475, 3521, 3814, 4324, 4529, 4801, 5086, 5149, 6225, 6726, 6829, 7374, 8225, 8464, 8775, 9454, 9601, 13926, 15625, 15876, 16129, 16900, 17161
OFFSET
1,3
COMMENTS
Trivially includes powers of 25, since 25^k = 100..00_5 = 10^(2k) when read in base 10. Moreover, for any a(n) in the sequence, 25*a(n) is also in the sequence. One could call "primitive" the terms not of this form, these would be 1, 4, 36 = 121_5, 49 = 144_5, 89 = 324_5, ... These primitive terms include the subsequence 25^k + 2*5^k + 1 = (5^k+1)^2, k > 0, which yields A033934 when written in base 5.
MATHEMATICA
Select[Range[0, 17200], IntegerQ@ Sqrt@ FromDigits@ IntegerDigits[#, 5] &] (* Michael De Vlieger, Jan 24 2016 *)
PROG
(PARI) is(n, b=5, c=10)=issquare(subst(Pol(digits(n, b)), x, c))
(Python)
A267765_list = [int(d, 5) for d in (str(i**2) for i in range(10**6)) if max(d) < '5'] # Chai Wah Wu, Mar 12 2016
CROSSREFS
Cf. A267763 - A267769 for bases 3 through 9. The base-2 analog is A000302 = powers of 4.
Sequence in context: A370575 A240164 A169600 * A376157 A231176 A335350
KEYWORD
nonn,base
AUTHOR
M. F. Hasler, Jan 20 2016
STATUS
approved