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

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, 15, 12, 14, 14, 13, 11, 13, 15, 15, 14, 15, 20, 14, 15, 19, 20, 16, 17, 17, 17, 21, 24, 23, 24, 28, 23, 25, 24, 27, 25, 32, 29, 25, 21, 26, 31, 31, 29, 36, 32

Offset: 1

Author

Michel Lagneau, Mar 23 2015

Keywords

• nonn

Comments

Conjecture: a(n)>0 for n>5.

Note that a(n) = A241535(n) = A241536(n) = 0 for n=1,2 and 5. - Michel Marcus, Mar 26 2015

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).

Examples

			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.
		

Links

• Michel Lagneau, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A001358, A071681, A241535, A241536.

Cf. A064911, A000040.

Programs

Haskell

a253138 n = sum $ map a064911 $
   takeWhile (> 0) $ map (2 * p -) $ dropWhile (< p) a001358_list
   where p = a000040 n
-- Reinhard Zumkeller, Mar 27 2015

Maple

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:

Mathematica

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 *)