A067760 a(n) is the least positive k such that (2n+1) + 2^k is prime, or 0 if no such k exists.

1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 3, 2, 1, 1, 4, 2, 1, 2, 1, 1, 2, 1, 5, 2, 1, 3, 2, 1, 1, 8, 2, 1, 2, 1, 1, 4, 2, 1, 2, 1, 7, 2, 1, 3, 4, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 7, 4, 5, 3, 4, 2, 1, 2, 1, 3, 2, 1, 1, 10, 3, 3, 2, 1, 1, 4, 2, 1, 4, 2, 1, 2, 1, 5, 2, 1, 3, 2, 1, 1, 4, 3, 3, 2, 1, 1, 2, 1, 1, 6, 5, 3, 6

Offset: 0

Author

Don Reble, Feb 05 2002

Keywords

• nonn

Comments

From Phil Moore (moorep(AT)lanecc.edu), Dec 14 2009: (Start)

It is known that a(39278) = 0, since no such prime exists for the Sierpiński number 78557 (cf. A076336).

It has recently been discovered that 2131+2^4583176 and 41693+2^5146295 are probable primes, so a(1065) is probably 4583176 and a(20846) is probably 5146295.

At present, the only odd value less than 78557 for which no prime or strong probable prime of the form t+2^k is known is t = 40291, so a(20145) is completely unknown. In addition, for 25 values of t < 78557, only strong probable primes are known. (End)

The last case was resolved in 2011 when the probable prime 40291+2^9092392 was found as a part of a distributed project "Five or Bust". See links. - Jeppe Stig Nielsen, Mar 29 2019

Examples

			a(15)=4 because (2*15+1)+2^k is composite for k=1,2,3 and prime for k=4.
		

Links

• Jinyuan Wang, Table of n, a(n) for n = 0..39278 (terms 0..1064 from T. D. Noe, terms 1065..3000 from Richard N. Smith).
• Mersenne Forum, Five or Bust
• Carlos Rivera, Puzzle 167. Primes m + 2^j & m - 2^j, The Prime Puzzles and Problems Connection.

Crossrefs

Cf. A016014, A050412, A066081, A033919, A094076, A076336, A260350, A263874, A263875 (records).

Programs

PARI

a(n) = {my(k=1); while (! isprime((2*n+1)+2^k), k++); k;} \\ Michel Marcus, Feb 26 2018