A173140 Numbers k such that the period of the decimal representation of 1/k is a square.

3, 9, 17, 51, 81, 101, 153, 163, 187, 257, 303, 323, 489, 561, 577, 729, 771, 883, 909, 969, 1111, 1241, 1297, 1377, 1467, 1683, 1717, 1731, 1801, 1919, 2261, 2313, 2329, 2649, 2771, 2827, 2907, 2997, 3137, 3333, 3349, 3529, 3553, 3667, 3723, 3891, 4039, 4199, 4267, 4369, 4401, 4883

Offset: 1

Author

Michel Lagneau, Feb 10 2010

Keywords

• nonn
• base
• changed

Comments

The period p of 1/k is given by the smallest integer p for which 10^p == 1 (mod k). The periods of unit fractions are given by sequence A002329.

Examples

			period of 1/3 = 1;
period of 1/9 = 1;
period of 1/17 = 16;
period of 1/51 = 16.
		

References

• J. W. L. Glaisher, On circulating decimals, Proc. Camb. Phil. Soc., 3 (1878), 185-206.
• D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-12.

Links

• Eric Weisstein's World of Mathematics, Decimal Expansion
• Eric Weisstein's World of Mathematics, Multiplicative Order

Programs

Maple

with(numtheory):n0:=60: ii:=1:tabl:=array(1..n0+1): for n from 2 to 10000 do: for p from 1 to 10000 while(irem(10^p,n)<>1 or gcd(n,10)<> 1) do: od: if irem(10^p,n) = 1 and gcd(n,10) = 1 and sqrt(p) = floor(sqrt(p)) then tabl[ii]:=n:ii:=ii+1:else fi:od: print(tabl):

Extensions

Title rephrased, more terms and Maple program added; corrected by T. D. Noe and edited by Michel Lagneau, Apr 26 2010