Sum Of (m/k), Over Some Divisors k

[Leroy Quet]

An idea for a family of sequences:

Sometimes interesting sequences result if we sum, not over k (k being the 
positive diuvisors of m), but over the (m/k)'s.

For example, if we take

a(m) = sum{k|m,k=prime power>1} (m/k),

we get (warning: figured by hand):

0, 1, 1, 3, 1, 5, 1, 7, 4, 7, 1, 13,...

(Also, a(m) = sum{j|m} phi(m/j) b(j), where b(j) is the sum of the 
prime-factorization exponents of j.)

(Summing over just the primes gets sequence A069359.)



And 2 interesting recursively generated sequences are:

(Again, sequences figured by hand.)


c(1) = 1;
c(m) = sum{k|m, k is among {c(1),c(2),...,c(m-1)}}  (m/k)

1, 2, 3, 6, 5, 12, 7, 12, 12, 17, 11, 25,...


f(1) = 1;

f(m) = sum{k|m, k not among {f(1),f(2),...,f(m-1)}}  (m/k)

1, 1, 1, 3, 1,  4, 1, 5, 1,  6, 1, 7,...


If we replace the (m/k) with k in the sequences {c(m)} and {f(m)}, 
however, then we would simply get the relatively uninteresting sequences 
{1,1,1,1,...} and {1,2,3,4,5,6,...}.

Neither {a(m)}, {c(m)}, nor {f(m)} is in the EIS, it seems (if I 
calculated by hand correctly).

thanks,
Leroy Quet