A329293 Number of positive integers k such that A002805(k) is not divisible by n, or a(n) = 0 if there are infinitely many such numbers.

0, 1, 11, 3, 19, 11, 97, 7, 35, 19

Offset: 1

Author

Jinyuan Wang, Dec 27 2019

Keywords

• nonn
• more
• hard

Comments

There are two cases where a(n) = 0: (a) n divides A002805(k) for all k, which only happens for n = 1; (b) there are infinitely many k such that n does not divide A002805(k), which may happen for some primes p and their multiples.

For prime p and k >= p, A002805(k) is not divisible by p if and only if p divides A001008(floor(k/p)), which means a(p) mod p = p - 1.

If k == -1 or 0 (mod p), then p divides A001008(k) iff p^2 divides A001008(floor(k/p)), otherwise p divides A001008(k) iff p divides the numerator of (Sum_{i=floor(k/p)*p+1..k} 1/i) + (Sum_{i=1..floor(k/p)} 1/i)/p, where p is an odd prime and k >= p. See A329061 for more information.

Links

• Jinyuan Wang, PARI program and examples for n <= 10
• Eric Weisstein's World of Mathematics, Harmonic Number

Crossrefs

Cf. A001008, A002805, A329061.