1,1

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. [From T. D. Noe (noe(AT)sspectra.com), Mar 30 2009]

Michel Waldschmidt, Open Diophantine problems [From T. D. Noe (noe(AT)sspectra.com), Mar 30 2009]

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.

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

A158245 (primitive terms), A158912 (four-digit repdigit numbers) [From T. D. Noe (noe(AT)sspectra.com), Mar 30 2009]

Sequence in context: A160843 A153368 A068600 this_sequence A158245 A076851 A164576

Adjacent sequences: A158232 A158233 A158234 this_sequence A158236 A158237 A158238

nice,nonn

T. D. Noe (noe(AT)sspectra.com), Mar 14 2009

Edited the inequality T. D. Noe (noe(AT)sspectra.com), Mar 30 2009