

A073761


Primitive numbers whose decimal expansion of 1/n is equidistributed in base 10.


2



61, 131, 181, 461, 491, 541, 571, 701, 811, 821, 941, 971, 1021, 1051, 1091, 1171, 1181, 1291, 1301, 1349, 1381, 1531, 1571, 1621, 1741, 1811, 1829, 1861, 2141, 2221, 2251, 2341, 2371, 2411, 2621, 2731, 2741, 2851, 2861, 2971, 3011, 3221, 3251, 3301
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OFFSET

1,1


COMMENTS

Usually once a number has the desired property, so do all its multiples. However there are exceptions. 61*7 in base 10 is not equidistributed. Multiples of earlier numbers are not included here.
There are 58 composite terms below 100000, 2 among which being even: a(239) = 25064 = 2^3 * 13 * 241, and a(613) = 72728 = 2^3 * 9091.
Conjecture 1: let p be a prime such that ord(10,p) is a multiple of 10, where ord(a,m) denotes the multiplicative order of a modulo m. Then p is a term if and only if 10 is a primitive root modulo p.
Conjecture 2: suppose that m is a term with bigomega(m) = 2, then m = p*q, where p == 1 (mod 10), q == 9 (mod 10), gcd(p1,q1) = 2, ord(10,p) = (p1)/2, and ord(10,q) = q1. Note that the converse is not true, though.
There are no counterexamples to the conjectures above below 100000.
Is there any odd term m such that bigomega(m) > 2? (End)


REFERENCES

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, revised edition, London, England, 1997, entry 61, page 110.


LINKS



EXAMPLE

61 is a term because 1/61 = .016393... (period 60 digits, 6 of each 0,1,..9).


MATHEMATICA

a = {}; Do[d = RealDigits[1/n][[1, 1]]; If[ !IntegerQ[d] && Count[d, 0] == Count[d, 1] == Count[d, 2] == Count[d, 3] == Count[d, 4] == Count[d, 5] == Count[d, 6] == Count[d, 7] == Count[d, 8] == Count[d, 9], If[ Select[n/a, IntegerQ] == {}, a = Append[a, n]]], {n, 11, 3330}]; a


CROSSREFS



KEYWORD

nonn,base


AUTHOR



EXTENSIONS



STATUS

approved



