A308249 Squares of automorphic numbers in base 12 (cf. A201918).

0, 1, 16, 81, 4096, 6561, 263169, 1478656, 40960000, 205549569, 54988374016, 233605955584, 6263292059649, 303894740860929, 338531738189824, 170196776412774400, 709858175909625856, 18638643564726714369, 124592287100855910400, 2576097707358918017025, 479214351668445504864256

Offset: 1

Author

Jeremias M. Gomes, May 17 2019

Keywords

• nonn
• base

Comments

(sqrt(m))12 is a suffix of m_12. - _A.H.M. Smeets, Aug 09 2019

All terms k^2 in this sequence (except the trivials 0 and 1) have a square root k that is the suffix of one of the 12-adic numbers given by A259468 or A259469. From this, the sequence has an infinite number of terms. - A.H.M. Smeets, Aug 09 2019

Examples

			4096 = 2454_12 and sqrt(2454_12) = 54_12. Hence 4096 is in the sequence.
		

Links

• V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.

Crossrefs

Cf. A201918, A259468, A259469.

Programs

Python

dig = "0123456789AB"
def To12(n):
    s = ""
    while n > 0:
        s, n = dig[n%12]+s, n//12
    return s
n, m = 1, 0
print(n,m*m)
while n < 100:
    m = m+1
    m2, m1 = To12(m*m), To12(m)
    i, i2, i1 = 0, len(m2), len(m1)
    while i < i1 and (m2[i2-i-1] == m1[i1-i-1]):
        i = i+1
    if i == i1:
        print(n,m*m)
n = n+1 # A.H.M. Smeets, Aug 09 2019

Sage

[(n * n) for n in (0..1000000) if (n * n).str(base = 12).endswith(n.str(base = 12))]

Formula

Equals A201918(n)^2.

Extensions

Terms a(16)..a(21) from A.H.M. Smeets, Aug 09 2019