A075226 Largest prime in the numerator of the 2^n sums generated from the set 1, 1/2, 1/3,..., 1/n.

3, 11, 19, 137, 137, 1019, 2143, 7129, 7129, 78167, 81401, 1085933, 1111673, 1165727, 2364487, 41325407, 41325407, 796326437, 809074601, 812400209, 822981689, 19174119571, 19652175721, 99554817251, 100483070801

Offset: 2

Author

T. D. Noe, Sep 08 2002

Keywords

• nice
• nonn

Comments

For the smallest odd prime not generated, see A075227. For information about how often the numerator of these sums is prime, see A075188 and A075189. The Mathematica program also prints the subset that yields the largest prime. For n <=20, the largest prime occurs in a sum of n-2, n-1, or n reciprocals.

Examples

			a(3) =11 because 11 is largest prime numerator in the three sums that yield primes: 1+1/2 = 3/2, 1/2+1/3 = 5/6 and 1+1/2+1/3 = 11/6.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 2..441 (terms 2..100 from Martin Fuller)
• Martin Fuller, PARI program

Crossrefs

Cf. A001008, A075135, A075188, A075189, A075227, A010051, A217712.

Programs

Haskell

import Data.Ratio (numerator)
a075226 n = a075226_list !! (n-1)
a075226_list = f 2 [recip 1] where
   f x hs = (maximum $ filter ((== 1) . a010051') (map numerator hs')) :
            f (x + 1) hs' where hs' = hs ++ map (+ recip x) hs
-- Reinhard Zumkeller, May 28 2013

Mathematica

Needs["DiscreteMath`Combinatorica`"]; maxN=20; For[t={}; lst={}; mx=0; i=0; n=2, n<=maxN, n++, While[i<2^n-1, i++; s=NthSubset[i, Range[n]]; k=Numerator[Plus@@(1/s)]; If[PrimeQ[k], If[k>mx, t=s]; mx=Max[mx, k]]]; Print[n, " ", t]; AppendTo[lst, mx]]; lst
Table[Max[Select[Numerator[Total/@Subsets[1/Range[n],{2,2^n}]],PrimeQ]],{n,2,30}] (* The program will take a long time to run. *) (* Harvey P. Dale, Jan 08 2019 *)

PARI

See Fuller link.

Python

from math import gcd, lcm
from itertools import combinations
from sympy import isprime
def A075226(n):
    m = lcm(*range(1,n+1))
    c, mlist = 0, tuple(m//i for i in range(1,n+1))
    for l in range(n,-1,-1):
        if sum(mlist[:l]) < c:
            break
        for p in combinations(mlist,l):
            s = sum(p)
            s //= gcd(s,m)
            if s > c and isprime(s):
                c = s
    return c # Chai Wah Wu, Feb 14 2022

Extensions

More terms from Martin Fuller, Jan 19 2008