A308249
Squares of automorphic numbers in base 12 (cf. A201918).
Original entry on oeis.org
0, 1, 16, 81, 4096, 6561, 263169, 1478656, 40960000, 205549569, 54988374016, 233605955584, 6263292059649, 303894740860929, 338531738189824, 170196776412774400, 709858175909625856, 18638643564726714369, 124592287100855910400, 2576097707358918017025, 479214351668445504864256
Offset: 1
4096 = 2454_12 and sqrt(2454_12) = 54_12. Hence 4096 is in the sequence.
-
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
-
[(n * n) for n in (0..1000000) if (n * n).str(base = 12).endswith(n.str(base = 12))]
A237583
Automorphic numbers: n^2 ends with n in base 6.
Original entry on oeis.org
0, 1, 3, 4, 9, 28, 81, 136, 1216, 6561, 16768, 29889, 76545, 203392, 636417, 1043200, 3995649, 6082048, 24151041, 36315136, 326481921, 689278977, 1487503360, 11573190657, 76876660737, 155240824833, 314944159744, 785129144320, 2035980763137, 4857090670593
Offset: 1
From A201821:
a(3) = (3)_6 = 3 since 3^2 = 9 = (13)_6 ends with 3 in base 6.
a(4) = (4)_6 = 4 since 4^2 = 16 = (24)_6 ends with 4 in base 6.
a(5) = (13)_6 = 9 since 9^2 = 81 = (213)_6 ends with 13 in base 6.
A201821
Automorphic numbers: n^2 ends with n in base 6 (written in base 6).
Original entry on oeis.org
0, 1, 3, 4, 13, 44, 213, 344, 5344, 50213, 205344, 350213, 1350213, 4205344, 21350213, 34205344, 221350213, 334205344, 2221350213, 3334205344, 52221350213, 152221350213, 403334205344, 5152221350213, 55152221350213, 155152221350213, 400403334205344
Offset: 1
a(3) = (3)_6 = 3 since 3^2 = 9 = (13)_6 ends with 3 in base 6.
a(4) = (4)_6 = 4 since 4^2 = 16 = (24)_6 ends with 4 in base 6.
a(5) = (13)_6 = 9 since 9^2 = 81 = (213)_6 ends with 13 in base 6.
- Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, p. 147.
- Maurice Kraitchik, Mathematical Recreations, New York, Dover, (2nd ed.) 1953, p. 77.
A201919
Automorphic numbers n^2 ends with n in base 14 (written in base 10).
Original entry on oeis.org
0, 1, 7, 8, 49, 148, 344, 2401, 36016, 151264, 386561, 1764736, 5764801, 46941952, 58471553, 374712065, 1101076992, 4802079233, 15858967552, 139825248256, 149429406721, 1595702681601, 2453862488064, 14602557997056, 42091354378241, 127990382747648
Offset: 1
a(3) = 7 = (7)_14 since 7^2 = 49 = (37)_14 ends with 7 in base 14.
a(4) = 8 = (8)_14 since 8^2 = 64 = (48)_14 ends with 8 in base 14.
a(5) = 49 = (37)_14 since 49^2 = 2401 = (C37)_14 ends with 37 in base 14.
A201921
Automorphic numbers: n^2 ends with n in base 15 (written in base 10).
Original entry on oeis.org
0, 1, 6, 10, 100, 126, 1000, 2376, 4375, 46251, 156250, 603126, 3640626, 7750000, 19140625, 151718751, 835156251, 1727734375, 5960937501, 32482421875, 236621093751, 340029296875, 8413134765625, 60784912109376, 68961425781250, 709516601562501, 1236678466796875
Offset: 1
a(3) = 6 = (6)_15 since 6^2 = 36 = (26)_15 ends with 6 in base 15.
a(4) = 10 = (A)_15 since 10^2 = 100 = (6A)_15 ends with A in base 15.
a(5) = 100 = (6A)_15 since 100^2 = 10000 = (2E6A)_15 ends with 6A in base 15.
A201948
Automorphic numbers: n^2 ends with n in base 18 (written in base 10).
Original entry on oeis.org
0, 1, 9, 10, 81, 244, 729, 5104, 6561, 98416, 413344, 1476225, 9034497, 24977728, 263063296, 349156737, 2711943424, 8308017153, 96467701761, 101891588608, 1286623443969, 2283843782656, 30847581595648, 33420828483585, 352189631991808, 804641749434369
Offset: 1
a(3) = 9 = (9)_18 since 9^2 = 81 = (49)_18 ends with 9 in base 18.
a(4) = 10 = (A)_18 since 10^2 = 100 = (5A)_18 ends with A in base 18.
a(5) = 81 = (49)_18 since 81^2 = 6561 = (1249)_18 ends with 49 in base 18.
Showing 1-6 of 6 results.
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