|
Search: id:A070824
|
|
|
| A070824 |
|
Number of divisors of n>=2 which are >1 and <n (nontrivial divisors). |
|
+0
4
|
|
| 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 4, 0, 2, 2, 3, 0, 4, 0, 4, 2, 2, 0, 6, 1, 2, 2, 4, 0, 6, 0, 4, 2, 2, 2, 7, 0, 2, 2, 6, 0, 6, 0, 4, 4, 2, 0, 8, 1, 4, 2, 4, 0, 6, 2, 6, 2, 2, 0, 10, 0, 2, 4, 5, 2, 6, 0, 4, 2, 6, 0, 10, 0, 2, 4, 4, 2, 6, 0, 8, 3
(list; graph; listen)
|
|
|
OFFSET
|
2,5
|
|
|
COMMENT
|
Sometimes mistakenly called proper divisors (see A032741)
a(n) = number of ordered factorizations of n into two factors, n = 2,3,... If n has the prime factorization n=Product p^e(j), j=1..r, the number of compositions of the vector (e(1), ..., e(r)) equals the number of ordered factorizations of n. Andrews (1998, page 59) gives a formula for the number of m-compositions of (e(1), ..., e(r)) which equals the number f(n,m) of ordered m-factorizations of n. But with m=2 the formula reduces to f(n,2)=d(n)-2=a(n). - A. O. Munagi (amunagi(AT)yahoo.com), Mar 31 2005
|
|
REFERENCES
|
Andrews, G. E., The Theory of Partitions, Addison-Wesley, Reading 1976; reprinted, Cambridge University Press, Cambridge, 1984, 1998.
|
|
FORMULA
|
a(n)=A000005(n)-2, n>=2 (with the divisor function d(n)=A000005(n)).
a(n) = d(n)-2, where d(n) is the divisor function. E.g. a(12)=4 because 12 has 4 ordered factorizations into two factors: 2*6, 6*2, 3*4, 4*3. - A. O. Munagi (amunagi(AT)yahoo.com), Mar 31 2005
|
|
EXAMPLE
|
a(12)=4 with the nontrivial divisors 2,3,4,6.
|
|
MAPLE
|
seq(numtheory[tau](n)-2, n=1..100); (Munagi)
|
|
CROSSREFS
|
Cf. A074206, A032741.
Sequence in context: A068067 A046926 A074398 this_sequence A071459 A070288 A117929
Adjacent sequences: A070821 A070822 A070823 this_sequence A070825 A070826 A070827
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), May 08, 2002
|
|
|
Search completed in 0.002 seconds
|
