A333196 Least k such that Sum_{i=1..n} k^n / i is a positive integer.

1, 2, 6, 6, 30, 10, 70, 70, 210, 210, 2310, 2310, 30030, 30030, 30030, 30030, 510510, 510510, 9699690, 1939938, 646646, 646646, 14872858, 44618574, 223092870, 223092870, 223092870, 223092870, 6469693230, 6469693230, 200560490130, 200560490130, 18232771830

Offset: 1

Author

Jinyuan Wang, Mar 10 2020

Keywords

• nonn

Comments

Least k > 0 such that k^n/A002805(n) is an integer.

Examples

			For n = 6, the denominator of Sum_{i=1..6} 1/i is 20 = 2^2*5, so a(7) = 2*5 = 10.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..2370

Crossrefs

Cf. A007947, A001008, A002805, A330030, A333072.

Programs

Maple

a:= n-> mul(i[1], i=ifactors(denom(harmonic(n)))[2]):
seq(a(n), n=1..33);  # Alois P. Heinz, Apr 23 2025

PARI

a(n) = factorback(factorint(denominator(sum(i=2, n, 1/i)))[, 1]);

Python

from functools import reduce
from operator import mul
from sympy import harmonic, factorint
def A333196(n):
    fs = factorint(harmonic(n).q)
    return 1 if len(fs) == 0 else reduce(mul,(p**(fs[p]//n + 1 if fs[p] % n else fs[p]//n) for p in fs)) # Chai Wah Wu, Apr 03 2020

Formula

a(n) = A007947(A002805(n)).