A267765 Numbers whose base-5 representation is a square when read in base 10.

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.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

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.

For a "prime" analog see also A235265, A235266, A152079, A235461 - A235482, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924.

Sequence in context: A370575 A240164 A169600 * A231176 A335350 A199772

Adjacent sequences: A267762 A267763 A267764 * A267766 A267767 A267768

Keyword

nonn,base

Author

M. F. Hasler, Jan 20 2016

Status

approved