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A253138 Number of ways to represent the n-th prime as the arithmetic mean of two semiprimes.

%I A253138 #23 Mar 27 2015 18:08:55
%S A253138 0,0,1,1,0,1,1,1,1,2,1,3,2,4,3,4,3,3,5,7,6,5,5,8,8,7,9,7,10,10,12,11,
%T A253138 15,12,14,14,13,11,13,15,15,14,15,20,14,15,19,20,16,17,17,17,21,24,23,
%U A253138 24,28,23,25,24,27,25,32,29,25,21,26,31,31,29,36,32
%N A253138 Number of ways to represent the n-th prime as the arithmetic mean of two semiprimes.
%C A253138 Conjecture: a(n)>0 for n>5.
%C A253138 Note that a(n) = A241535(n) = A241536(n) = 0 for n=1,2 and 5. - _Michel Marcus_, Mar 26 2015
%C A253138 Among the a(n) decompositions of prime(n) into two semiprimes (prime(n)+ k)/2 and (prime(n)-k)/2, there is one where k is minimum with k = A241536(n) and there is one where k is maximum with k = prime(n) - A241535(n).
%H A253138 Michel Lagneau, <a href="/A253138/b253138.txt">Table of n, a(n) for n = 1..1000</a>
%e A253138 a(12)=3 as prime(12) = 37 = (9+65)/2 = (25+49)/2 =(35+39)/2 where 9, 25, 35, 39, 49 and 65 are semiprime.
%p A253138 with(numtheory):for n from 1 to 100 do:c:=0:p:=ithprime(n):for m from 1 to p-1 do:p1:=p-m:p2:=p+m:if bigomega(p1)=2 and bigomega(p2)=2 then c:=c+1:else fi:od:printf(`%d, `,c):od:
%t A253138 Reap[For[n=1, n <= 100, n++, c=0; p = Prime[n]; For[m=1, m <= p-1, m++, p1 = p-m; p2 = p+m; If[PrimeOmega[p1] == 2 && PrimeOmega[p2] == 2 , c = c+1]]; Print[c]; Sow[c]]][[2, 1]] (* _Jean-François Alcover_, Mar 23 2015, translated from Maple *)
%o A253138 (Haskell)
%o A253138 a253138 n = sum $ map a064911 $
%o A253138    takeWhile (> 0) $ map (2 * p -) $ dropWhile (< p) a001358_list
%o A253138    where p = a000040 n
%o A253138 -- _Reinhard Zumkeller_, Mar 27 2015
%Y A253138 Cf. A001358, A071681, A241535, A241536.
%Y A253138 Cf. A064911, A000040.
%K A253138 nonn
%O A253138 1,10
%A A253138 _Michel Lagneau_, Mar 23 2015