OFFSET
1,2
COMMENTS
Sum of n-th row of triangleA210208.[Reinhard Zumkeller,Mar 18 2012]
LINKS
Reinhard Zumkeller,Table of n, a(n) for n = 1..10000
FORMULA
a(1) = 1, a(p) = p+1, a(pq) = p+q+1, a(pq...z) = (p+q+...+z) + 1, a(p^k) = (p^(k+1)-1) / (p-1), for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q,..., z.
G.f.: x/(1 - x) + Sum_{k>=2} floor(1/ Omega (k))*k*x^k/(1 - x^k), where Omega (k) is the number of distinct prime factors (A001221). -Ilya Gutkovskiy,Jan 04 2017
EXAMPLE
For n = 12, set of such divisors is {1, 2, 3, 4}; a(12) = 1+2+3+4 = 10. From
MAPLE
f:= n -> 1 + add((t[1]^(t[2]+1)-t[1])/(t[1]-1), t=ifactors(n)[2]):
map(f, [$1..100]); #Robert Israel,Jan 04 2017
MATHEMATICA
Array[ Plus @@ (Select[ Divisors[ # ], (Length[ FactorInteger[ # ] ]<=1)& ])&, 70 ]
PROG
(PARI) for(n=1, 100, s=1; fordiv(n, d, if((ispower(d,, &z)&&isprime(z)) || isprime(d), s+=d)); print1(s, "," ))
(Haskell)
a023888 = sum. a210208_row --Reinhard Zumkeller,Mar 18 2012
(PARI)
a(n) = {
my(f = factor(n), fsz = matsize(f)[1]);
1 + sum(k = 1, fsz, f[k, 1]*(f[k, 1]^f[k, 2] - 1)\(f[k, 1]-1));
};
vector(100, n, a(n)) \\Gheorghe Coserea,Jan 04 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved