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

1,3

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

Links

Table of n, a(n) for n=1..21.

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

Formula

Equals A201918(n)^2.

Example

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

Prog

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

Crossrefs

Cf. A201918, A259468, A259469.

Sequence in context: A053909 A151502 A030693 * A285989 A231303 A218082

Adjacent sequences: A308246 A308247 A308248 * A308250 A308251 A308252

Keyword

nonn,base

Author

Jeremias M. Gomes, May 17 2019

Extensions

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

Status

approved