OFFSET
1,8
COMMENTS
a(n) is the number of divisor pairs of n, (d1,d2), such that d1<d2 and where d1,d2 have the same number of distinct prime factors. For example, a(8) = 3 since we have the ordered pairs (2,4), (2,8) and (4,8), where the divisors in each pair have the same number of distinct prime factors.
FORMULA
a(n) >= 1 if and only if n is composite. (E.g. composite numbers either have a divisor pair of the form (p,p^k), where p is prime and k is a positive integer > 1, which implies that omega(p) = omega(p^k) = 1, or they have a divisor pair of the form (p,q) where p and q are distinct primes and omega(p) = omega(q) = 1. Then the total number of such divisor pairs is >= 1.)
Furthermore, a(n) = 0 if and only if n is noncomposite. (E.g. a(1) = 0 since 1 has no divisor pairs such that d1<d2, and a(p) = 0 (for p prime) since the only divisor pair of p such that d1<d2 is (1,p), of which, omega(1) = 0 but omega(p) = 1. So the primes have no such divisor pairs.)
MATHEMATICA
Table[Sum[Sum[KroneckerDelta[PrimeNu[i], PrimeNu[k]] (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k - 1}], {k, n}], {n, 100}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Aug 30 2020
STATUS
approved