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A253138
Number of ways to represent the n-th prime as the arithmetic mean of two semiprimes.
2
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
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
KEYWORD
nonn
AUTHOR
Michel Lagneau, Mar 23 2015
STATUS
approved