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A158235 Numbers n whose square can be represented as a repdigit number in some base < n. 6
11, 20, 39, 40, 49, 78, 133, 247, 494, 543, 1086, 1218, 1629, 1651, 1729, 2172, 2289, 2715, 3097, 3258, 3458, 3801, 4171, 4344, 4503, 4578, 4887, 5187, 5430, 6194, 6231, 6867, 6916, 7303, 7540, 7563, 8342, 8645, 8773, 9139, 9156, 9291, 10374, 12103 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Alternatively, numbers n such that n^2 = d*(b^k-1)/(b-1) for some b, d, and k with d<b<n.

It appears that 11^2 and 20^2 are the only squares representable as repunits having more than two "digits" in some base. Some bases, such as 313, appear many times. Why? Can a square have more than one representation? See A158236 for the bases and A158237 for the repdigit. The representations of 11^2, 20^2, 40^2, and 1218^2 have more than 3 "digits". Is the list of such numbers finite?

A generalization of this problem, to "determine all perfect powers with identical digits in some basis", is briefly mentioned on page 6 of Waldschmidt's paper. - T. D. Noe, Mar 30 2009

LINKS

Table of n, a(n) for n=1..44.

Michel Waldschmidt, Open Diophantine problems, arXiv:math/0312440 [math.NT], 2003-2004.

EXAMPLE

11^2 = 11111 in base 3.

20^2 = 1111 in base 7.

39^2 = 333 in base 22.

40^2 = 4444 in base 7.

49^2 = 777 in base 18.

78^2 = (12)(12)(12) in base 22.

1218^2 = (21)(21)(21)(21) in base 41.

MATHEMATICA

Do[sq=n^2; Do[If[Length[Union[IntegerDigits[sq, b]]]==1, Print[{n, sq, b, IntegerDigits[sq, b]}]], {b, 2, n}], {n, 10000}]

CROSSREFS

Cf. A158245 (primitive terms), A158912 (four-digit repdigit numbers).

Sequence in context: A160843 A153368 A068600 * A158245 A076851 A164576

Adjacent sequences:  A158232 A158233 A158234 * A158236 A158237 A158238

KEYWORD

nice,nonn

AUTHOR

T. D. Noe, Mar 14 2009

EXTENSIONS

Inequality edited by T. D. Noe, Mar 30 2009

STATUS

approved

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Last modified August 19 07:12 EDT 2017. Contains 290794 sequences.