A267768 Numbers whose base-8 representation is a square when read in base 10.

0, 1, 4, 14, 21, 30, 52, 64, 81, 100, 149, 174, 212, 241, 256, 289, 382, 405, 446, 532, 622, 661, 804, 849, 896, 1012, 1045, 1102, 1220, 1281, 1344, 1409, 1476, 1557, 1630, 1780, 1920, 2001, 2197, 2286, 2452, 2545, 2593, 2878, 2965, 3070, 3233, 3328, 3441, 3540, 3630, 3733, 4068, 4096

Offset: 1

Author

M. F. Hasler, Jan 20 2016

Keywords

• nonn
• base

Comments

Trivially includes powers of 64, since 64^k = 100..00_8 = 10^(2k) when read in base 10. Moreover, for any a(n) in the sequence, 64*a(n) is also in the sequence. One could call "primitive" the terms not of this form. These primitive terms include the subsequence 64^k + 2*8^k + 1 = (8^k+1)^2, k > 0, which yields A033934 when written in base 8.

Motivated by the subsequence A267490 which lists the primes in this sequence.

Links

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

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.

Programs

Magma

[n: n in [0..10^4] | IsSquare(Seqint(Intseq(n, 8)))]; // Vincenzo Librandi, Dec 28 2016

Mathematica

Select[Range[0, 2 10^4], IntegerQ@Sqrt@FromDigits@IntegerDigits[#, 8] &] (* Vincenzo Librandi, Dec 28 2016 *)

PARI

is(n,b=8,c=10)=issquare(subst(Pol(digits(n,b)),x,c))

Python

A267768_list = [int(s, 8) for s in (str(i**2) for i in range(10**6)) if max(s) < '8'] # Chai Wah Wu, Jan 20 2016