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A308249 Squares of automorphic numbers in base 12 (cf. A201918). 0
0, 1, 16, 81, 4096, 6561, 263169, 1478656, 40960000, 205549569, 54988374016, 233605955584, 6263292059649, 303894740860929, 338531738189824, 170196776412774400, 709858175909625856, 18638643564726714369, 124592287100855910400, 2576097707358918017025, 479214351668445504864256 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 23 16:20 EDT 2021. Contains 348215 sequences. (Running on oeis4.)