A337525 a(n) = Sum_{d1|n, d2|n, d1<d2} [omega(d1) = omega(d2)], where omega is the number of distinct prime factors of n (A001221) and [ ] is the Iverson bracket.

0, 0, 0, 1, 0, 1, 0, 3, 1, 1, 0, 4, 0, 1, 1, 6, 0, 4, 0, 4, 1, 1, 0, 9, 1, 1, 3, 4, 0, 6, 0, 10, 1, 1, 1, 12, 0, 1, 1, 9, 0, 6, 0, 4, 4, 1, 0, 16, 1, 4, 1, 4, 0, 9, 1, 9, 1, 1, 0, 17, 0, 1, 4, 15, 1, 6, 0, 4, 1, 6, 0, 25, 0, 1, 4, 4, 1, 6, 0, 16, 6, 1, 0, 17, 1, 1, 1, 9, 0, 17, 1, 4

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.

Links

Table of n, a(n) for n=1..92.

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

Cf. A001221 (omega).

Sequence in context: A058395 A035694 A006941 * A076277 A130115 A191582

Adjacent sequences: A337522 A337523 A337524 * A337526 A337527 A337528

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 30 2020

Status

approved