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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. 1

%I #11 Jan 23 2023 12:37:25

%S 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 54,57,59,60,62,80,80,80,82,84,101,101,102,102,104,110,119,122,123,

%U 124,125,133,136,138,138,153,154,158,159,160,167,174,174,178,178,182,185,189,189

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

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

%F a(n+1) - a(n) = A027833(n) - 2, n > 1. [_R. J. Mathar_, Aug 29 2009]

%e a(1)=1 counts the isolated prime 2, which smaller than 3;

%e a(2)=1 counts the isolated prime 2, which is smaller than 5;

%e a(5)=2 counts the isolated primes 2 and 23, which are smaller than 29;

%e a(6)=3 counts 2, 23 and 37, which are smaller than 41.

%p read("transforms3") ; tp := BFILETOLIST("b001359.txt") ;

%p 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 seq(A162309(n),n=1..130 ); # _R. J. Mathar_, Aug 29 2009

%t A027833 =

%t Differences[Flatten[Position[Differences[Prime[Range[500]]], 2]]];

%t ReplacePart[Accumulate[Join[{2}, A027833 - 2]], 1 -> 1]

%t (* _Jean-François Alcover_, Jan 23 2023, after _Harvey P. Dale_ in A027833 *)

%Y Cf. A000040, A141468, A073425.

%K nonn

%O 1,5

%A _Juri-Stepan Gerasimov_, Jul 01 2009

%E 53 replaced with 54, 100 removed twice, etc., by _R. J. Mathar_, Aug 29 2009

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)