A101288 The number of primes between the n-th isolated prime and n-th isolated composite.

1, 5, 6, 7, 5, 5, 4, 2, 2, 3, 3, 4, 9, 6, 7, 10, 12, 10, 12, 13, 15, 26, 27, 30, 36, 41, 43, 46, 48, 49, 68, 69, 70, 73, 76, 94, 95, 97, 98, 97, 104, 114, 118, 118, 120, 122, 131, 135, 138, 139, 153, 155, 160, 162, 162, 170, 178, 177, 182, 181, 184, 188, 191, 192, 194

Offset

1,2

Comments

Instead of "isolated" one speaks also of "single" primes, they are listed in A007510 and include 2 by convention. By isolated composites are meant composites whose two neighbors both are prime, i.e., the averages of twin primes, A014574. - M. F. Hasler, Aug 11 2015

Links

Table of n, a(n) for n=1..65.

Example

a(1) = 1 = # { 3 }, the only prime between 2 (the first "isolated prime" according to A007510) and 4, the first "isolated composite" in the sense that x-1 and x+1 both are primes.
a(2) = 5 = # { 7, 11, 13, 17, 19 }, the primes between the second isolated prime, 23, and second isolated composite, 6.
a(3) = 6 = # { 31, 29, 23, 19, 17, 13 }, the primes between A007510(3) = 37 and A014574(3) = 12.

Maple

A007510 := proc(n) if n = 1 then 2; else for a from procname(n-1)+1 do if isA007510(a) then return a; end if; end do; end if; end proc:
isA007510 := proc(n) isprime(n) and not isprime(n+2) and not isprime(n-2) ; simplify(%) ; end proc:
A101288 := proc(n) if n = 1 then return 1 ; end if; a7510 := A007510(n) ; a4574 := A014574(n) ; if a7510 > a4574 then numtheory[pi](a7510-1)-numtheory[pi](a4574) ; else numtheory[pi](a4574)-numtheory[pi](a7510+1) ; end if; end proc:
seq(A101288(n), n=1..120) ; # R. J. Mathar, Apr 25 2010

Crossrefs

Cf. A000040, A007510, A014574.

Sequence in context: A342218 A299082 A171423 * A212479 A385599 A095942

Adjacent sequences: A101285 A101286 A101287 * A101289 A101290 A101291

Keyword

nonn

Author

Juri-Stepan Gerasimov, Nov 02 2009

Extensions

More terms from R. J. Mathar, Apr 25 2010

Edited by Jon E. Schoenfield and M. F. Hasler, Aug 11 2015

Status

approved