A040017 Prime 3 followed by unique period primes (the period r of 1/p is not shared with any other prime) of the form A019328(r)/gcd(A019328(r),r) in order (periods r are given in A051627).
3, 11, 37, 101, 9091, 9901, 333667, 909091, 99990001, 999999000001, 9999999900000001, 909090909090909091, 1111111111111111111, 11111111111111111111111, 900900900900990990990991, 909090909090909090909090909091
Offset: 1
Examples
The decimal expansion of 1/101 is 0.00990099..., having a period of 4 and it is the only prime with that period.
References
- J.-P. Delahaye, Merveilleux nombres premiers ("Amazing primes"), p. 324, Pour la Science Paris 2000.
Links
- Robert G. Wilson v, Table of n, a(n) for n = 1..47
- Chris Caldwell, The Prime Glossary, Unique prime.
- C. K. Caldwell, "Top Twenty" page, Unique.
- Chris K. Caldwell and Harvey Dubner, Unique-Period Primes, J. Recreational Math., 29:1 (1998) 43-48.
- Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
- Eric Weisstein's World of Mathematics, Unique Prime.
- Wikipedia, Unique prime.
- Index entries for sequences related to decimal expansion of 1/n
Programs
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Mathematica
lst = {}; Do[c = Cyclotomic[n, 10]; q = c/GCD[c, n]; If[PrimeQ[q], AppendTo[lst, q]], {n, 62}]; Prepend[Sort[lst], 3] (* Arkadiusz Wesolowski, May 13 2012 *)
Formula
For n >= 2, a(n) = A019328(r) / gcd(A019328(r), r), where r = A051627(n). - Max Alekseyev, Oct 14 2022
Extensions
Missing term a(45) inserted in b-file at the suggestion of Eric Chen by Max Alekseyev, Oct 13 2022
Edited by Max Alekseyev, Oct 14 2022
Comments