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%I #81 Feb 26 2024 17:08:56
%S 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,
%T 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,
%U 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
%N a(n) = n * Sum_{p|n} 1/p where p are primes dividing n.
%C 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
%C a(n) = n-1 iff n = 1 or n is a primary pseudoperfect number A054377. - _Jonathan Sondow_, Apr 16 2014
%C a(1) = 0 by the standard convention for empty sums.
%C “Seva” on the MathOverflow link asks if the iterates of this sequence are all eventually 0. - _Charles R Greathouse IV_, Feb 15 2019
%H Antti Karttunen, <a href="/A069359/b069359.txt">Table of n, a(n) for n = 1..16384</a> (first 10000 terms from Franklin T. Adams-Watters)
%H Antti Karttunen, <a href="/A069359/a069359.txt">Data supplement: n, a(n) computed for n = 1..100000</a>
%H MathOverflow, <a href="https://mathoverflow.net/q/323194/">A recursion with a number-theoretic function</a> (2019)
%H Joshua Zelinsky, <a href="https://arxiv.org/abs/2402.14234">The sum of the reciprocals of the prime divisors of an odd perfect or odd primitive non-deficient number</a>, arXiv:2402.14234 [math.NT], 2024.
%F 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
%F 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
%F a(A054377(n)) = A054377(n) - 1. - _Jonathan Sondow_, Apr 16 2014
%F Dirichlet g.f.: zeta(s - 1)*primezeta(s). - _Geoffrey Critzer_, Mar 17 2015
%F Sum_{k=1..n} a(k) ~ A085548 * n^2 / 2. - _Vaclav Kotesovec_, Feb 04 2019
%F From _Antti Karttunen_, Nov 15 2019: (Start)
%F a(n) = Sum_{d|n} A008683(n/d)*A323599(d).
%F a(n) = A003415(n) - A329039(n) = A230593(n) - n = A306369(n) - A000010(n).
%F a(n) = A276085(A329350(n)) = A048675(A329352(n)).
%F a(A276086(n)) = A329029(n), a(A328571(n)) = A329031(n).
%F (End)
%F a(n) = Sum_{d|n} A000010(d) * A001221(n/d). - _Torlach Rush_, Jan 21 2020
%F a(n) = Sum_{k=1..n} omega(gcd(n, k)). - _Ilya Gutkovskiy_, Feb 21 2020
%e 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
%p A069359 := n -> add(n/d, d = select(isprime, numtheory[divisors](n))):
%p seq(A069359(i), i = 1..20); # _Peter Luschny_, Jan 31 2012
%p # second Maple program:
%p a:= n-> n*add(1/i[1], i=ifactors(n)[2]):
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Oct 23 2019
%t f[list_, i_] := list[[i]]; nn = 100; a = Table[n, {n, 1, nn}]; b =
%t 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 *)
%o (Sage)
%o def A069359(n) :
%o D = filter(is_prime, divisors(n))
%o return add(n/d for d in D)
%o print([A069359(i) for i in (1..20)]) # _Peter Luschny_, Jan 31 2012
%o (PARI) a(n) = n*sumdiv(n, d, isprime(d)/d); \\ _Michel Marcus_, Mar 18 2015
%o (PARI) a(n) = my(ps=factor(n)[,1]~);sum(k=1,#ps,n\ps[k]) \\ _Franklin T. Adams-Watters_, Apr 09 2015
%o (Magma) [0] cat [n*&+[1/p: p in PrimeDivisors(n)]:n in [2..80]]; // _Marius A. Burtea_, Jan 21 2020
%o (Python)
%o from sympy import primefactors
%o def A069359(n): return sum(n//p for p in primefactors(n)) # _Chai Wah Wu_, Feb 05 2022
%Y Cf. A003415, A005117, A068328, A010051, A000027, A054377, A180253, A230593, A292786, A306369, A326690, A329029, A329350, A329352.
%Y Cf. A322068 (partial sums), A323599 (Inverse Möbius transform).
%Y 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).
%K nonn
%O 1,4
%A _Benoit Cloitre_, Apr 15 2002
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