A238797 Smallest k such that 2^k - (2*n+1) and (2*n+1)*2^k - 1 are both prime, k <= 2*n+1, or 0 if no such k exists.

0, 3, 4, 0, 0, 0, 0, 5, 6, 5, 7, 6, 9, 5, 0, 7, 6, 6, 0, 0, 10, 0, 6, 0, 7, 9, 6, 7, 8, 0, 17, 8, 0, 0, 7, 0, 0, 18, 0, 0, 0, 8, 0, 10, 8, 9, 18, 0, 0, 7, 0, 0, 8, 12, 0, 7, 0, 11, 16, 0, 21, 0, 0, 0, 8, 14, 0, 0, 18, 9, 10, 8, 77, 0, 0, 0, 12, 8, 0, 11, 18, 0

Offset: 0

Author

Ilya Lopatin and Juri-Stepan Gerasimov, Mar 05 2014

Keywords

• nonn

Comments

Numbers n such that 2^k - (2*n+1) and (2*n+1)*2^k - 1 are both prime:

For k = 0: 2, 3, 5, 7, 13, 17, ... Intersection of A000043 and A000043

for k = 1: 3, 4, 6, 94, ... Intersection of A050414 and A002235

for k = 2: 4, 8, 10, 12, 18, 32, ... Intersection of A059608 and A001770

for k = 3: Intersection of A059609 and A001771

for k = 4: 21, ... Intersection of A059610 and A002236

for k = 5: Intersection of A096817 and A001772

for k = 6: Intersection of A096818 and A001773

for k = 7: 5, 10, 14, ... Intersection of A059612 and A002237

for k = 8: 6, 16, 20, 36, ... Intersection of A059611 and A001774

for k = 9: 5, 21, ... Intersection of A096819 and A001775

for k = 10: 7, 13, ... Intersection of A096820 and A002238

for k = 11: 6, 8, 12, ...

for k = 12: 9, ...

for k = 13: 5, 8, 10, ...

Examples

			a(1) = 3 because 2^3 - (2*1+1) = 5 and (2*1+1)*2^3 - 1 = 23 are both prime, 3 = 2*1+1,
a(2) = 4 because 2^4 - (2*2+1) = 11 and (2*2+1)*2^4 - 1 = 79 are both prime, 4 < 2*2+1 = 5.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A238748, A238904 (smallest k such that 2^k + (2n+1) and (2n+1)*2^k + 1 are both prime, k <= n, or -1 if no such k exists).

Programs

Mathematica

a[n_] := Catch@ Block[{k = 1}, While[k <= 2*n+1, If[2^k - (2*n + 1) > 0 && PrimeQ[2^k - (2*n+1)] && PrimeQ[(2*n + 1)*2^k-1], Throw@k]; k++]; 0]; a/@ Range[0, 80] (* Giovanni Resta, Mar 15 2014 *)

Extensions

a(0), a(19), a(20) corrected by Giovanni Resta, Mar 13 2014