A125503 Smallest number k such that the numerator of the generalized harmonic number H(k,n) = Sum_{i=1..k} 1/i^n is a prime.

2, 2, 3, 2, 23, 73, 15, 2, 3, 5, 13, 57, 3, 171, 5, 2, 21, 7, 55, 8902, 26, 1298, 115, 139, 3, 2019, 3, 4, 3, 15, 56, 177

Offset: 1

Author

Alexander Adamchuk, Dec 28 2006, Jan 31 2007

Keywords

• nonn
• hard
• more

Comments

a(n) = 2 for n = {1,2,4,8,16,...}. Corresponding Fermat primes A019434.

a(n) = 3 for n = {3,9,13,25,27,29,95,107,153,159,...}.

a(n) = 5 for n = {10,15,60,90,197,209,...}.

a(n) = 7 for n = {18,47,112,155,273,...}.

a(n) = 15 for n = {7,30,43,...}.

a(21) = 26. a(28) = 4. a(31) = 56. a(144) = 9.

From Alexander Adamchuk, Apr 18 2010: (Start)

a(22)-a(25) = {1298,115,139,3}.

a(27)-a(32) = {3,4,3,15,56,177}.

a(n) = 3 for all n>2 listed in A125706. (End)

a(26) = 2019. - Alexander Adamchuk, Apr 26 2010

a(20) > 3000. - Michael S. Branicky, Jun 25 2022

Links

• Eric Weisstein's World of Mathematics, Harmonic Number

Crossrefs

Cf. A001008, A007406, A007408, A007410, A099828, A019434, A125706.

Programs

Mathematica

Do[n = 1; f = 0; While[Not[PrimeQ[Numerator[f]]], f = f + 1/n^x; n++ ]; Print[{x, n - 1}], {x, 1, 25}] (* Alexander Adamchuk, Apr 18 2010 *)

PARI

a(n) = my(k=1); while (!ispseudoprime(numerator(sum(i=1, k, 1/i^n))), k++); k; \\ Michel Marcus, Jun 04 2022

Python

from sympy import isprime
from fractions import Fraction
def a(n):
    Hkn, k = Fraction(1, 1), 1
    while not isprime(Hkn.numerator):
        k += 1
        Hkn += Fraction(1, k**n)
    return k
print([a(n) for n in range(1, 20)]) # Michael S. Branicky, Jun 11 2022

Extensions

a(22)-a(25) from Alexander Adamchuk, Apr 18 2010

a(26)-a(32) from Alexander Adamchuk, Apr 26 2010

Incorrect a(20) removed by Michael S. Branicky, Jun 25 2022

a(20) = 8902 from Michael S. Branicky, Jun 12 2023