A252168 Smallest k > 0 such that |(2n-1) - 2^k| is prime, or -1 if no such k exists.

2, 3, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 1, 1, 2, 3, 1, 2, 1, 1, 2, 1, 2, 4, 1, 2, 4, 1, 1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 4, 1, 2, 4, 3, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 3, 4, 4, 47, 1, 2, 1, 2, 6, 1, 1, 2, 3, 3, 8, 1, 1, 2, 3, 1, 2, 5, 1, 2, 1, 2, 4

Offset: 1

Author

Eric Chen, Dec 14 2014

Keywords

• nonn
• hard

Comments

It is known that a(254602) = -1, because |509203-2^k| is always divisible by 3, 5, 7, 13, 17, or 241. a(1147) is the first unknown term.

a((A101036(n)+1)/2) = -1, so there are infinitely many n such that a(n) = -1.

a((A133122(n)+1)/2) = A096502((A133122(n)-1)/2).

Examples

			a(12) = 2 because 2*12-1 = 23 and that 23-2^1 = 21 is not prime but 23-2^2 = 19 is.
a(69) = 6 because 2*69-1 = 137, |137-2^k| is composite for k = 1, 2, 3, 4, 5 and prime for k = 6.
Even the smallest k can be also very large. For example, a(169) = 791.
a(1147) > 65536.
		

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..1146

Crossrefs

Cf. A006285, A096502, A133122, A188903.

Cf. A046067, A046069, A067760.

Programs

Mathematica

Table[k = 1; While[!PrimeQ[Abs[(2*n-1) - 2^k]], k++]; k, {n, 1, 1000}]

PARI

A252168(n)={ my(k=1); n=2*n-1; while(!ispseudoprime(abs(n-2^k)), k++); k }

Extensions

a(19) corrected by Jinyuan Wang, Mar 25 2023