Super-3 Number

An Integer $n$ such that $3n^3$ contains three consecutive 3s in its Decimal representation. The first few super-3 numbers are 261, 462, 471, 481, 558, 753, 1036, ... (Sloane's A014569). A. Anderson has conjectured that all numbers ending in 471, 4710, or 47100 are super-3 (Pickover 1995).

For a digit $d$, super-3 numbers can be generalized to super-$d$ numbers $n$ such that $dn^d$ contains $d$ $d$s in its Decimal representation. The following table gives the first few super-$d$ numbers for small $d$.

$d$	Sloane	Super-$d$ numbers
2	Sloane's A032743	19, 31, 69, 81, 105, 106, 107, 119, 127, ...
3	Sloane's A014569	261, 462, 471, 481, 558, 753, 1036, 1046, ...
4	Sloane's A032744	1168, 4972, 7423, 7752, 8431, 10267, 11317, ...
5	Sloane's A032745	4602, 5517, 7539, 12955, 14555, 20137, 20379, ...
6	Sloane's A032746	27257, 272570, 302693, 323576, 364509, 502785, ...
7	Sloane's A032747	140997, 490996, 1184321, 1259609, 1409970, ...
8	Sloane's A032748	185423, 641519, 1551728, 1854230, 6415190, ...
9	Sloane's A032749	17546133, 32613656, 93568867, 107225764, ...

References

Pickover, C. A. Keys to Infinity. New York: Wiley, p. 7, 1995.

Sloane, N. J. A. Sequence A014569 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.