A064560 Numbers n such that reciprocal of n terminates with an infinite repetition of digit 1. Multiples of 10 are omitted.

9, 576, 36864, 140625, 2359296, 150994944, 2197265625, 9663676416, 618475290624, 34332275390625, 39582418599936, 2533274790395904, 162129586585337856, 536441802978515625, 10376293541461622784, 664082786653543858176

Offset

1,1

References

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 73-82.

Links

Table of n, a(n) for n=1..16.

Index entries for sequences related to decimal expansion of 1/n

Formula

These are the numbers 9*2^(6i) and 9*5^(6i). Proof: We have 1/n = (k+ (1/9))/ 10^m so n = 9*10^m/(9k+1) so 9 divides n, say n=9x and 10 does not divide x. Then x = 10^m/(9k+1), so 9k+1 = 2^r 5^s (say) and m = max(r, s). Also 9 | 2^r 5^s - 1 (*). But phi(9) = 6, 2^6 == 1 mod 9, 5^6 == 1 mod 9 so only solution to (*) is r == s mod 6. If r >= s, say r = s + 6i then n = 9*5^(6i). Similarly if s >= r, n = 9*2^(6i). - N. J. A. Sloane, Sep 21 2001

Example

1/36864 = 0.0000271267361111111111...

Prog

(ARIBAS): a064560(1, 200000000, 1, 36). Definition of a064560: function a064560(minarg, maxarg, d, len: integer); var n, r, decpos, sign: integer; s, sd: string; begin set_floatprec(round(len*3.3)); r := len div 3; sd := alloc(string, r, ftoa(d)[0]); n := minarg - 1; while n < maxarg do inc(n); if n mod 10 <> 0 then s := float_ecvt(1/n, len, decpos, sign); if s[len-r-1..len-2] = sd then write(n, " "); end; end; end; end;

Crossrefs

Cf. A064561-A064567.

Sequence in context: A347844 A291910 A074731 * A264121 A095968 A233067

Adjacent sequences: A064557 A064558 A064559 * A064561 A064562 A064563

Keyword

nonn,base,easy

Author

Patrick De Geest, Sep 20 2001

Status

approved