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
Keywords
Examples
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.
Programs
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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 *)
Formula
a(n+1) - a(n) = A027833(n) - 2, n > 1. [R. J. Mathar, Aug 29 2009]
Extensions
53 replaced with 54, 100 removed twice, etc., by R. J. Mathar, Aug 29 2009
Comments