A039669 Numbers n > 2 such that n - 2^k is a prime for all k > 0 with 2^k < n.
4, 7, 15, 21, 45, 75, 105
Offset: 1
Examples
45 is here because 43, 41, 37, 29 and 13 are primes.
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
- P. Erdős, On integers of the form 2^k + p and some related questions, Summa Bras. Math., 2 (1950), 113-123.
- Walter E. Mientka and Roger C. Weitzenkamp, On f-plentiful numbers, Journal of Combinatorial Theory, Volume 7, Issue 4, December 1969, pages 374-377.
Crossrefs
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).
Cf. A022005.
Programs
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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
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Mathematica
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 *) -
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
Extensions
Additional comments from T. D. Noe, Sep 15 2002
Definition edited by Robert Israel, Dec 23 2015
Comments