A069359 a(n) = n * Sum_{p|n} 1/p where p are primes dividing n.

0, 1, 1, 2, 1, 5, 1, 4, 3, 7, 1, 10, 1, 9, 8, 8, 1, 15, 1, 14, 10, 13, 1, 20, 5, 15, 9, 18, 1, 31, 1, 16, 14, 19, 12, 30, 1, 21, 16, 28, 1, 41, 1, 26, 24, 25, 1, 40, 7, 35, 20, 30, 1, 45, 16, 36, 22, 31, 1, 62, 1, 33, 30, 32, 18, 61, 1, 38, 26, 59, 1, 60, 1, 39, 40, 42, 18, 71, 1, 56

Offset: 1

Author

Benoit Cloitre, Apr 15 2002

Keywords

• nonn

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

Examples

			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
		

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. See also Integers (2025) Vol. 25, Art. No. A59, page 2.

Crossrefs

Cf. A003415, A005117, A068328, A010051, A000027, A054377, A180253, A230593, A292786, A306369, A326690, A329029, A329350, A329352.

Cf. A322068 (partial sums), A323599 (Inverse Möbius transform).

Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), this sequence (k=1), A322078 (k=2), A351242 (k=3), A351244 (k=4), A351245 (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).

Programs

Magma

[0] cat [n*&+[1/p: p in PrimeDivisors(n)]:n in [2..80]]; // Marius A. Burtea, Jan 21 2020

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 *)

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

Python

from sympy import primefactors
def A069359(n): return sum(n//p for p in primefactors(n)) # Chai Wah Wu, Feb 05 2022

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

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

a(A054377(n)) = A054377(n) - 1. - Jonathan Sondow, Apr 16 2014

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)

a(n) = Sum_{d|n} A008683(n/d)*A323599(d).

a(n) = A003415(n) - A329039(n) = A230593(n) - n = A306369(n) - A000010(n).

a(n) = A276085(A329350(n)) = A048675(A329352(n)).

a(A276086(n)) = A329029(n), a(A328571(n)) = A329031(n).

(End)

a(n) = Sum_{d|n} A000010(d) * A001221(n/d). - Torlach Rush, Jan 21 2020

a(n) = Sum_{k=1..n} omega(gcd(n, k)). - Ilya Gutkovskiy, Feb 21 2020

a(p^k) = p^(k-1) for p prime and k>=1. - Wesley Ivan Hurt, Jul 15 2025