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

%I A162309 #11 Jan 23 2023 12:37:25
%S A162309 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,
%T A162309 54,57,59,60,62,80,80,80,82,84,101,101,102,102,104,110,119,122,123,
%U A162309 124,125,133,136,138,138,153,154,158,159,160,167,174,174,178,178,182,185,189,189
%N A162309 a(n) is the number of isolated primes up to the smaller component of the n-th twin prime pair.
%C A162309 Cardinality of the set of terms of A007510 which are smaller than A001359(n).
%F A162309 a(n+1) - a(n) = A027833(n) - 2, n > 1. [_R. J. Mathar_, Aug 29 2009]
%e A162309 a(1)=1 counts the isolated prime 2, which smaller than 3;
%e A162309 a(2)=1 counts the isolated prime 2, which is smaller than 5;
%e A162309 a(5)=2 counts the isolated primes 2 and 23, which are smaller than 29;
%e A162309 a(6)=3 counts 2, 23 and 37, which are smaller than 41.
%p A162309 read("transforms3") ; tp := BFILETOLIST("b001359.txt") ;
%p A162309 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:
%p A162309 seq(A162309(n),n=1..130 ); # _R. J. Mathar_, Aug 29 2009
%t A162309 A027833 =
%t A162309 Differences[Flatten[Position[Differences[Prime[Range[500]]], 2]]];
%t A162309 ReplacePart[Accumulate[Join[{2}, A027833 - 2]], 1 -> 1]
%t A162309 (* _Jean-François Alcover_, Jan 23 2023, after _Harvey P. Dale_ in A027833 *)
%Y A162309 Cf. A000040, A141468, A073425.
%K A162309 nonn
%O A162309 1,5
%A A162309 _Juri-Stepan Gerasimov_, Jul 01 2009
%E A162309 53 replaced with 54, 100 removed twice, etc., by _R. J. Mathar_, Aug 29 2009