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%I #48 Nov 12 2014 09:36:05
%S 1217,1249,1553,4049,4273,4481,4993,5297,6449,6481,6689,7121,8081,
%T 8609,9137,9281,10337,10369,10433,11617,11633,12577,13441,13633,14321,
%U 14753,15569,16417,16433,16673,17137,18257,18433,18481,19793,20113,20353,23057,23857
%N Primes p such that A179382((p+1)/2) = (p-1)/16.
%C Let n be a natural number coprime to 10 and let c be the "cycle length of n" (defined below).
%C Conjecture 1: If n-1=2^x*c for some x<5, then n is prime. If x > 4, the relative density of primes in such numbers is 1.
%C Conjecture 2: If the period of the decimal expansion of 1/n is n-1 or a divisor of n-1, and if n-1=2^x*c or n+1=2^x*c for some x, then n is prime.
%C - Lear Young, with contributions from Peter Košinár, Giovanni Resta, Charles R Greathouse IV, May 22 2013
%C To define the "cycle length of n" (using n=73 as an example):
%C Step 1 : 73 + 1 = 74. Get the odd part of 74, which is 37
%C Step 2 : 73 + 37 = 110. Get the odd part of 110, which is 55
%C Step 3 : 73 + 55 = 128. Get the odd part of 128, which is 1
%C Continuing this operation (with 73+1) repeats the same steps as above. There are 3 steps in the cycle, so the cycle length of 73 is c=3. (same to A179382((73+1)/2)=3).
%C More for the "cycle length of n" see link and cross references.
%C The numbers in the sequence are the values of n in the above conjecture when c=4 in case (1).
%H Lear Young and Charles R Greathouse IV, <a href="/A225759/b225759.txt">Table of n, a(n) for n = 1..10000</a> (first 117 terms from Young)
%H Hagen von Eitzen, <a href="http://math.stackexchange.com/questions/394408/how-to-prove-these-two-ways-give-the-same-numbers">Details of the "cycle length of n"</a>
%e (1217-1)/16 = 76 = A179382(609).
%o (PARI) oddres(n)=n>>valuation(n, 2)
%o cyc(d)=my(k=1, t=1); while((t=oddres(t+d))>1, k++); k
%o forstep(n=17,1e4,[32,16],if(cyc(n)==n>>4 && isprime(n), print1(n", ")))
%o \\ _Charles R Greathouse IV_, May 15 2013
%Y Analogs with different values of c: A001122 when c=1, A155072 when c=2, A001134 when c=3. A225890 has composite values.
%Y Cf. A179382, A136042 (both sequences related to the way to get the "cycle length of n").
%K nonn
%O 1,1
%A _Lear Young_, May 15 2013
%E Edited by _Charles R Greathouse IV_, Nov 11 2014
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