A162309 a(n) is the number of isolated primes up to the smaller component of the n-th twin prime pair.

1, 1, 1, 1, 2, 3, 5, 6, 10, 10, 13, 13, 17, 17, 17, 19, 20, 23, 24, 26, 29, 39, 39, 43, 50, 54, 57, 59, 60, 62, 80, 80, 80, 82, 84, 101, 101, 102, 102, 104, 110, 119, 122, 123, 124, 125, 133, 136, 138, 138, 153, 154, 158, 159, 160, 167, 174, 174, 178, 178, 182, 185, 189, 189

Offset

1,5

Comments

Cardinality of the set of terms of A007510 which are smaller than A001359(n).

Links

Table of n, a(n) for n=1..64.

Formula

a(n+1) - a(n) = A027833(n) - 2, n > 1. [R. J. Mathar, Aug 29 2009]

Example

a(1)=1 counts the isolated prime 2, which smaller than 3;
a(2)=1 counts the isolated prime 2, which is smaller than 5;
a(5)=2 counts the isolated primes 2 and 23, which are smaller than 29;
a(6)=3 counts 2, 23 and 37, which are smaller than 41.

Maple

read("transforms3") ; tp := BFILETOLIST("b001359.txt") ;
A162309 := proc(n) global tp; a := 0 ; for j from 2 to op(n, tp)-1 do if isprime(j) then if ( j in tp ) or (j-2) in tp then ; else a :=a +1; fi; fi; od: a ; end:
seq(A162309(n), n=1..130 ); # R. J. Mathar, Aug 29 2009

Mathematica

A027833 =
Differences[Flatten[Position[Differences[Prime[Range[500]]], 2]]];
ReplacePart[Accumulate[Join[{2}, A027833 - 2]], 1 -> 1]
(* Jean-François Alcover, Jan 23 2023, after Harvey P. Dale in A027833 *)

Crossrefs

Cf. A000040, A141468, A073425.

Sequence in context: A061939 A029503 A133139 * A014593 A261088 A303431

Adjacent sequences: A162306 A162307 A162308 * A162310 A162311 A162312

Keyword

nonn

Author

Juri-Stepan Gerasimov, Jul 01 2009

Extensions

53 replaced with 54, 100 removed twice, etc., by R. J. Mathar, Aug 29 2009

Status

approved