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%I #46 Mar 17 2023 12:58:33
%S 4,7,15,21,45,75,105
%N Numbers n > 2 such that n - 2^k is a prime for all k > 0 with 2^k < n.
%C Erdős conjectures that these are the only values of n with this property.
%C No other terms below 2^120. - _Max Alekseyev_, Dec 08 2011
%C Curiously, Mientka and Weitzenkamp say there are 9 such numbers below 20000. - _Michel Marcus_, May 12 2013
%C Presumably, Mientka and Weitzenkamp are including 1 and 2. - _Robert Israel_, Dec 23 2015
%C Observation: The prime numbers of the form (n-2) 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
%D R. K. Guy, Unsolved Problems in Number Theory, A19.
%D F. Le Lionnais, Les Nombres Remarquables, Paris, Hermann, 1983, p. 96, 1983.
%D D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 306.
%D D. Wells, Curious and interesting numbers, Penguin Books, p. 118.
%H P. Erdős, <a href="http://www.renyi.hu/~p_erdos/1950-07.pdf">On integers of the form 2^k + p and some related questions</a>, Summa Bras. Math., 2 (1950), 113-123.
%H Walter E. Mientka and Roger C. Weitzenkamp, <a href="http://dx.doi.org/10.1016/S0021-9800(69)80067-0">On f-plentiful numbers</a>, Journal of Combinatorial Theory, Volume 7, Issue 4, December 1969, pages 374-377.
%e 45 is here because 43, 41, 37, 29 and 13 are primes.
%t lst={}; Do[k=1; While[p=n-2^k; p>0 && PrimeQ[p], k++ ]; If[p<=0, AppendTo[lst, n]], {n, 3, 1000}]; lst (* _T. D. Noe_, Sep 15 2002 *)
%o (PARI) isok(n) = {my(k = 1); while (2^k < n, if (! isprime(n-2^k), return (0)); k++;); return (1);} \\ _Michel Marcus_, Dec 14 2015
%o (MATLAB)
%o N = 10^8; % to get terms < N
%o p = primes(N);
%o A = [3:N];
%o for k = 1:floor(log2(N))
%o A = intersect(A, [1:(2^k), (p+2^k)]);
%o end
%o A % _Robert Israel_, Dec 23 2015
%Y Cf. A067526 (n such that n-2^k is prime or 1), A067527 (n such that n-3^k is prime), A067528 (n such that n-4^k is prime or 1), A067529 (n such that n-5^k is prime), A100348 (n such that n-4^k is prime), A100349 (n such that n-2^k is prime or semiprime), A100350 (primes p such that p-2^k is prime or semiprime), A100351 (n such that n-2^k is semiprime).
%Y Cf. A022005.
%K nonn,hard,more
%O 1,1
%A _Felice Russo_
%E Additional comments from _T. D. Noe_, Sep 15 2002
%E Definition edited by _Robert Israel_, Dec 23 2015
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