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

1,10

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).

Links

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

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A001358, A071681, A241535, A241536.

Cf. A064911, A000040.

Sequence in context: A188901 A171618 A344096 * A283994 A026238 A066136

Adjacent sequences: A253135 A253136 A253137 * A253139 A253140 A253141

Keyword

nonn

Author

Michel Lagneau, Mar 23 2015

Status

approved