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 A192869 Thin primes: odd primes p such that p+1 is a prime (or 1) times a power of two. 7
 3, 5, 7, 11, 13, 19, 23, 31, 37, 43, 47, 61, 67, 73, 79, 103, 127, 151, 157, 163, 191, 193, 211, 223, 271, 277, 283, 313, 331, 367, 383, 397, 421, 457, 463, 487, 523, 541, 547, 607, 613, 631, 661, 673, 691, 733, 751, 757, 787, 823, 877, 907, 991, 997, 1051 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Broughan & Qizhi conjecture that a(n) << n (log n)^2, matching the lower bound they proved. Sequence A206581 excludes the Mersenne primes (A000043), which are included here under the "or 1" case. - T. D. Noe, Mar 07 2012 REFERENCES D. R. Heath-Brown, "Artin's conjecture for primitive roots", Quarterly Journal of Mathematics 37:1 (1986) pp. 27-38. N. M. Timofeev, "The Hardy-Ramanujan and Halasz inequalities for shifted primes", Mathematical Notes 57:5 (1995), pp. 522-535. LINKS T. D. Noe, Table of n, a(n) for n = 1..1000 Kevin Broughan and Zhou Qizhi, Flat primes and thin primes, Bulletin of the Australian Mathematical Society 82:2 (2010), pp. 282-292. Qizhi Zhou, Multiply perfect numbers of low abundancy, PhD thesis (2010) FORMULA a(n) >> n (log n)^2. MATHEMATICA onePrimeQ[n_] := n == 1 || PrimeQ[n]; Select[Prime[Range[2, 1000]], onePrimeQ[(# + 1)/2^IntegerExponent[# + 1, 2]] &] (* T. D. Noe, Mar 06 2012 *) PROG (PARI) is(n)=n%2&&isprime(n)&&(isprime((n+1)>>valuation(n+1, 2)) || n+1==1<

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