

A039669


Numbers n > 2 such that n  2^k is a prime for all k > 0 with 2^k < n.


16




OFFSET

1,1


COMMENTS

Erdős conjectures that these are the only values of n with this property.
No other terms below 2^120.  Max Alekseyev, Dec 08 2011
Curiously, Mientka and Weitzenkamp say there are 9 such numbers below 20000.  Michel Marcus, May 12 2013
Presumably, Mientka and Weitzenkamp are including 1 and 2.  Robert Israel, Dec 23 2015
Observation: The prime numbers of the form (n2) associated with each element of the series are (2,5,13,19,43,73,103). These prime numbers are exactly the first elements of A068374 (primes n such that positive values of n  A002110(k) are all primes for k>0).  David Morales Marciel, Dec 14 2015


REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, A19.
F. Le Lionnais, Les Nombres Remarquables, Paris, Hermann, 1983, p. 96, 1983.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 306.
D. Wells, Curious and interesting numbers, Penguin Books, p. 118.


LINKS

Table of n, a(n) for n=1..7.
P. Erdős, On integers of the form 2^k + p and some related questions, Summa Bras. Math., 2 (1950), 113123.
Walter E. Mientka and Roger C. Weitzenkamp, On fplentiful numbers, Journal of Combinatorial Theory, Volume 7, Issue 4, December 1969, pages 374377.


EXAMPLE

45 is here because 43, 41, 37, 29 and 13 are primes.


MATHEMATICA

lst={}; Do[k=1; While[p=n2^k; p>0 && PrimeQ[p], k++ ]; If[p<=0, AppendTo[lst, n]], {n, 3, 1000}]; lst (* T. D. Noe, Sep 15 2002 *)


PROG

(PARI) isok(n) = {my(k = 1); while (2^k < n, if (! isprime(n2^k), return (0)); k++; ); return (1); } \\ Michel Marcus, Dec 14 2015
(MATLAB)
N = 10^8; % to get terms < N
p = primes(N);
A = [3:N];
for k = 1:floor(log2(N))
A = intersect(A, [1:(2^k), (p+2^k)]);
end
A % Robert Israel, Dec 23 2015


CROSSREFS

Cf. A067526 (n such that n2^k is prime or 1), A067527 (n such that n3^k is prime), A067528 (n such that n4^k is prime or 1), A067529 (n such that n5^k is prime), A100348 (n such that n4^k is prime), A100349 (n such that n2^k is prime or semiprime), A100350 (primes p such that p2^k is prime or semiprime), A100351 (n such that n2^k is semiprime).
Sequence in context: A092309 A263617 A271675 * A109622 A269967 A124286
Adjacent sequences: A039666 A039667 A039668 * A039670 A039671 A039672


KEYWORD

nonn,hard,more


AUTHOR

Felice Russo


EXTENSIONS

Additional comments from T. D. Noe, Sep 15 2002
Definition edited by Robert Israel, Dec 23 2015


STATUS

approved



