A075188 Number of times that the numerator of a sum generated from the set 1, 1/2, 1/3,..., 1/n is prime.

0, 1, 3, 9, 19, 43, 79, 162, 307, 607, 1075, 2186, 3872, 7573, 15101, 29139, 52295, 104953, 189915, 379275, 754081, 1462115, 2675851, 5351541, 10254019, 19987942, 38901233, 77620568, 144021667, 288428481, 537642772, 1056802340, 2113152353, 4138261885

Offset: 1

Author

T. D. Noe, Sep 08 2002

Keywords

• nice
• nonn

Comments

Note that for each n there are only 2^(n-1) new sums to consider. Surprisingly, nearly half of the sums have a prime numerator. For the number of distinct primes, see A075189. For the largest generated prime, see A075226. For the smallest odd prime not generated, see A075227.

A217712(n) = number of primes occurring exactly once as numerators. - Reinhard Zumkeller, Jun 02 2013

Examples

			a(3) = 3 because 3 sums yield prime numerators: 1+1/2 = 3/2, 1/2+1/3 = 5/6 and 1+1/2+1/3 = 11/6.
		

Crossrefs

Cf. A001008, A075135, A075189, A075226, A075227.

Cf. A010051.

Programs

Haskell

import Data.Ratio (numerator)
a075188 n = a075188_list !! (n-1)
a075188_list = f 1 [] where
   f x hs = (length $ filter ((== 1) . a010051') (map numerator hs')) :
            f (x + 1) hs' where hs' = hs ++ map (+ recip x) (0 : hs)
-- Reinhard Zumkeller, May 28 2013

Mathematica

Needs["DiscreteMath`Combinatorica`"]; maxN=20; For[cnt=0; lst={}; i=0; n=1, n<=maxN, n++, While[i<2^n-1, i++; s=NthSubset[i, Range[n]]; k=Numerator[Plus@@(1/s)]; If[PrimeQ[k], cnt++ ]]; AppendTo[lst, cnt]]; lst

Extensions

a(21)-a(25) by Reinhard Zumkeller, May 28 2013

a(26)-a(31) from Chai Wah Wu, Feb 14 2022

a(32)-a(34) from Sean A. Irvine, Feb 10 2025