A163757 The count of primes between the n-th unsafe and the n-th safe prime.

1, 1, 0, 1, 6, 6, 11, 15, 25, 26, 32, 37, 49, 51, 54, 68, 67, 70, 76, 79, 98, 115, 118, 121, 132, 136, 159, 171, 176, 176, 178, 185, 192, 196, 210, 234, 244, 258, 258, 259, 264, 275, 308, 308, 318, 351, 357, 359, 365, 367, 370, 379, 382, 386, 418, 438, 455, 457, 462, 473, 477

Offset: 1

Author

Juri-Stepan Gerasimov, Aug 03 2009

Keywords

• nonn

Comments

For n=3, where the 3rd unsafe prime is larger than the 3rd safe prime, there are two primes in between which could formally be counted as -2, but have been replaced by 0 here.

Examples

			a(1)=1 counts one prime (the 3) between 2 and 5;
a(2)=1 counts one prime (the 5) between 3 and 7;
a(5)=6 counts the primes from 23 to 43 between 19 and 53.
		

Crossrefs

Cf. A000040, A005385, A059456.

Programs

Maple

isA005385 := proc(n) if isprime(n) then isprime( (n-1)/2 ) ; else false; fi; end:
isA059456 := proc(n) if isprime(n) then not isprime( (n-1)/2 ) ; else false; fi; end:
A059456 := proc(n) if n = 1 then 2; else for a from procname(n-1)+1 do if isA059456(a) then RETURN(a) ; fi; od: fi; end:
A005385 := proc(n) if n = 1 then 5; else for a from procname(n-1)+1 do if isA005385(a) then RETURN(a) ; fi; od: fi; end:
A000720 := proc(n) numtheory[pi](n) ; end:
A163757 := proc(n) max(0,A000720(A005385(n)-1)-A000720(A059456(n))) ; end: seq(A163757(n),n=1..80) ; # R. J. Mathar, Aug 06 2009

Formula

a(n) = max( 0, A000720(A005385(n)-1)-A000720(A059456(n)) ).

Extensions

Corrected by R. J. Mathar, Aug 06 2009