A217712 Number of primes occurring exactly once as numerators in sums generated from the set 1, 1/2, 1/3,..., 1/n.

0, 1, 3, 3, 11, 13, 27, 54, 106, 168, 378, 142, 733, 1597, 1283, 3418, 8204, 10112, 24644, 7829, 32866, 78136, 178741, 37002, 256392, 650596, 1402914, 286854, 2053463

Offset

1,3

Comments

For information about how often the numerator of the generated sums is prime, see A075188 and A075189; for the largest generated prime, see A075226; for the smallest odd prime not generated, see A075227.

Links

Table of n, a(n) for n=1..29.

Example

For n=3 there are the following fractions as sums of 1, 1/2 and 1/3:
{1/3, 1/2, 5/6, 1, 4/3, 3/2, 11/6}, three numerators are prime and they occur exactly once, therefore a(3) = A075188(3) = A075189(3) = #{3, 5, 11} = 3;
n=4: adding 1/4 to the previous fractions gives together: 1/4, 1/3, 1/2, 1/3+1/4=7/12, 1/2+1/4=3/4, 5/6, 1, 5/6+1/4=13/12, 1+1/4=5/4, 4/3, 3/2, 4/3+1/4=19/12, 3/2+1/4=7/4, 11/6 and 11/6+1/4=25/12:
A075188(4) = #{7/12, 3/4, 5/6, 13/12, 5/4, 3/2, 19/12, 7/4, 11/6} = 9,
A075189(4) = #{3, 5, 7, 11, 13, 19} = 6,
a(4) = #{11, 13, 19} = 3.

Prog

(Haskell)
import Data.Ratio ((%), numerator)
import Data.Set (Set, empty, fromList, toList, union, size)
import Data.Set (member, delete, insert)
a217712 n = a217712_list !! (n-1)
a217712_list = f 1 empty empty where
   f x s s1 = size s1' : f (x + 1) (s `union` fromList hs) s1' where
     s1' = g s1 $ filter ((== 1) . a010051') $ map numerator hs
     g v []                    = v
     g v (w:ws) | w `member` v = g (delete w v) ws
                | otherwise    = g (insert w v) ws
     hs = map (+ 1 % x) $ 0 : toList s

Crossrefs

Cf. A010051.

Sequence in context: A136123 A045495 A045494 * A321678 A027416 A281905

Adjacent sequences: A217709 A217710 A217711 * A217713 A217714 A217715

Keyword

nonn

Author

Reinhard Zumkeller, Jun 02 2013

Status

approved