OFFSET
1,4
COMMENTS
Coincides with arithmetic derivative on squarefree numbers: a(A005117(n)) = A068328(n) = A003415(A005117(n)). - Reinhard Zumkeller, Jul 20 2003, Clarified by Antti Karttunen, Nov 15 2019
a(n) = n-1 iff n = 1 or n is a primary pseudoperfect number A054377. - Jonathan Sondow, Apr 16 2014
a(1) = 0 by the standard convention for empty sums.
“Seva” on the MathOverflow link asks if the iterates of this sequence are all eventually 0. - Charles R Greathouse IV, Feb 15 2019
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..16384 (first 10000 terms from Franklin T. Adams-Watters)
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
MathOverflow, A recursion with a number-theoretic function (2019)
Joshua Zelinsky, The sum of the reciprocals of the prime divisors of an odd perfect or odd primitive non-deficient number, arXiv:2402.14234 [math.NT], 2024.
FORMULA
G.f.: Sum(x^p(j)/(1-x^p(j))^2,j>=1), where p(j) is the j-th prime. - Vladeta Jovovic, Mar 29 2006
a(n) = A230593(n) - n. a(n) = A010051(n) (*) A000027(n), where operation (*) denotes Dirichlet convolution, that is, convolution of type: a(n) = Sum_(d|n) b(d) * c(n/d) = Sum_(d|n) A010051(d) * A000027(n/d). - Jaroslav Krizek, Nov 07 2013
Dirichlet g.f.: zeta(s - 1)*primezeta(s). - Geoffrey Critzer, Mar 17 2015
Sum_{k=1..n} a(k) ~ A085548 * n^2 / 2. - Vaclav Kotesovec, Feb 04 2019
From Antti Karttunen, Nov 15 2019: (Start)
(End)
a(n) = Sum_{k=1..n} omega(gcd(n, k)). - Ilya Gutkovskiy, Feb 21 2020
EXAMPLE
a(12) = 10 because the prime divisors of 12 are 2 and 3 so we have: 12/2 + 12/3 = 6 + 4 = 10. - Geoffrey Critzer, Mar 17 2015
MAPLE
A069359 := n -> add(n/d, d = select(isprime, numtheory[divisors](n))):
seq(A069359(i), i = 1..20); # Peter Luschny, Jan 31 2012
# second Maple program:
a:= n-> n*add(1/i[1], i=ifactors(n)[2]):
seq(a(n), n=1..100); # Alois P. Heinz, Oct 23 2019
MATHEMATICA
f[list_, i_] := list[[i]]; nn = 100; a = Table[n, {n, 1, nn}]; b =
Table[If[PrimeQ[n], 1, 0], {n, 1, nn}]; Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Mar 17 2015 *)
PROG
(Sage)
def A069359(n) :
D = filter(is_prime, divisors(n))
return add(n/d for d in D)
print([A069359(i) for i in (1..20)]) # Peter Luschny, Jan 31 2012
(PARI) a(n) = n*sumdiv(n, d, isprime(d)/d); \\ Michel Marcus, Mar 18 2015
(PARI) a(n) = my(ps=factor(n)[, 1]~); sum(k=1, #ps, n\ps[k]) \\ Franklin T. Adams-Watters, Apr 09 2015
(Magma) [0] cat [n*&+[1/p: p in PrimeDivisors(n)]:n in [2..80]]; // Marius A. Burtea, Jan 21 2020
(Python)
from sympy import primefactors
def A069359(n): return sum(n//p for p in primefactors(n)) # Chai Wah Wu, Feb 05 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Apr 15 2002
STATUS
approved