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 A213025 Balanced semiprimes (of order one): semiprimes which are the average of the previous semiprime and the following semiprime. 4

%I

%S 34,86,94,122,142,185,194,202,214,218,262,289,302,314,321,358,371,394,

%T 407,413,415,422,446,471,489,493,497,517,535,562,581,586,626,634,669,

%U 687,698,734,785,791,815,838,842,922,982,989,1042,1057,1079,1135,1138

%N Balanced semiprimes (of order one): semiprimes which are the average of the previous semiprime and the following semiprime.

%C Semiprimes that are the average of three successive semiprimes.

%C First term not also in A086005 is 185. - _Alonso del Arte_, Jun 04 2012

%H Alois P. Heinz, <a href="/A213025/b213025.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Semiprime.html">Semiprime</a>

%F 2*sp_(n) = sp_(n - 1) + sp_(n + 1).

%F a(n) = 1/3 * (sp(i) + sp(i + 1) + sp(i + 2), for some i(n).

%e 194 is in the sequence because 194 = (187 + 194 + 201)/3 = (A001358(61) + A001358(62) + A001358(63))/3.

%p with(numtheory):

%p prevsp:= proc(n) local k; for k from n-1 by -1

%p while isprime(k) or bigomega(k)<>2 do od; k end:

%p nextsp:= proc(n) local k; for k from n+1

%p while isprime(k) or bigomega(k)<>2 do od; k end:

%p a:= proc(n) option remember; local s;

%p s:= `if`(n=1, 4, a(n-1));

%p do s:= nextsp(s);

%p if s=(prevsp(s)+nextsp(s))/2 then break fi

%p od; s

%p end:

%p seq (a(n), n=1..100); # _Alois P. Heinz_, Jun 03 2012

%t bspQ[{a_,b_,c_}]:=b==(a+c)/2; With[{sp=Select[Range[1200],PrimeOmega[#] == 2&]}, Transpose[Select[Partition[sp,3,1],bspQ]][[2]]] (* _Harvey P. Dale_, Nov 18 2012 *)

%o a213025 n = a213025_list !! (n-1)

%o a213025_list = f a001358_list where

%o f (x:sps'@(y:z:sps)) | 2 * y == (x + z) = y : f sps'

%o | otherwise = f sps'

%o -- _Reinhard Zumkeller_, Jun 10 2012

%Y Cf. A086005 (subsequence), A001358, A006562, A065516, A212820.

%K nonn

%O 1,1

%A _Gerasimov Sergey_, Jun 03 2012

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Last modified September 17 08:31 EDT 2019. Contains 327127 sequences. (Running on oeis4.)