A267769 Numbers whose base-9 representation is a square when read in base 10.

0, 1, 4, 15, 23, 33, 58, 73, 81, 100, 121, 185, 213, 265, 298, 324, 361, 400, 474, 509, 555, 643, 685, 751, 861, 914, 1093, 1153, 1215, 1288, 1354, 1481, 1554, 1705, 1783, 1863, 1945, 2029, 2210, 2301, 2488, 2584, 2673, 2773, 2875, 3101, 3210, 3424, 3538, 3682, 3802, 4038, 4154, 4281, 4450

Offset: 1

Author

M. F. Hasler, Jan 20 2016

Keywords

• nonn
• base

Comments

Trivially includes powers of 81, since 81^k = 100..00_9 = 10^(2k) when read in base 10. Moreover, for any a(n) in the sequence, 81*a(n) is also in the sequence. One could call "primitive" the terms not of this form. These primitive terms include the subsequence 81^k + 2*9^k + 1 = (9^k+1)^2, k > 0, which yields A033934 when written in base 9.

Links

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

Crossrefs

Cf. A267763 - A267768 for bases 3 through 8. 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.

Programs

Mathematica

Select[Range[0, 5000], IntegerQ@ Sqrt@ FromDigits@ IntegerDigits[#, 9] &] (* Michael De Vlieger, Jan 24 2016 *)

PARI

is(n,b=9,c=10)=issquare(subst(Pol(digits(n,b)),x,c))

Python

A267769_list = [int(s, 9) for s in (str(i**2) for i in range(10**6)) if max(s) < '9'] # Chai Wah Wu, Jan 20 2016