|
| |
|
|
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, 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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
|
|
|
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
|
|
|
|
STATUS
|
approved
|
| |
|
|
