A225759 Primes p such that A179382((p+1)/2) = (p-1)/16.

1217, 1249, 1553, 4049, 4273, 4481, 4993, 5297, 6449, 6481, 6689, 7121, 8081, 8609, 9137, 9281, 10337, 10369, 10433, 11617, 11633, 12577, 13441, 13633, 14321, 14753, 15569, 16417, 16433, 16673, 17137, 18257, 18433, 18481, 19793, 20113, 20353, 23057, 23857

Offset: 1

Author

Lear Young, May 15 2013

Keywords

• nonn

Comments

Let n be a natural number coprime to 10 and let c be the "cycle length of n" (defined below).

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.

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.

- Lear Young, with contributions from Peter Košinár, Giovanni Resta, Charles R Greathouse IV, May 22 2013

To define the "cycle length of n" (using n=73 as an example):

Step 1 : 73 + 1 = 74. Get the odd part of 74, which is 37

Step 2 : 73 + 37 = 110. Get the odd part of 110, which is 55

Step 3 : 73 + 55 = 128. Get the odd part of 128, which is 1

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).

More for the "cycle length of n" see link and cross references.

The numbers in the sequence are the values of n in the above conjecture when c=4 in case (1).

Examples

			(1217-1)/16 = 76 = A179382(609).
		

Links

• Lear Young and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 117 terms from Young)
• Hagen von Eitzen, Details of the "cycle length of n"

Crossrefs

Analogs with different values of c: A001122 when c=1, A155072 when c=2, A001134 when c=3. A225890 has composite values.

Cf. A179382, A136042 (both sequences related to the way to get the "cycle length of n").

Programs

PARI

oddres(n)=n>>valuation(n, 2)
cyc(d)=my(k=1, t=1); while((t=oddres(t+d))>1, k++); k
forstep(n=17,1e4,[32,16],if(cyc(n)==n>>4 && isprime(n), print1(n", ")))
\\ Charles R Greathouse IV, May 15 2013

Extensions

Edited by Charles R Greathouse IV, Nov 11 2014