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A039669 Numbers n > 2 such that n - 2^k is a prime for all k > 0 with 2^k < n. 16
4, 7, 15, 21, 45, 75, 105 (list; graph; refs; listen; history; text; internal format)
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 (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

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), 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.

EXAMPLE

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

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 *)

PROG

(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

(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 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).

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

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Last modified May 26 12:21 EDT 2017. Contains 287095 sequences.