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A069359 a(n) = n*sum(p|n,1/p) where p are primes dividing n. 15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Also arithmetic derivative of squarefree numbers: a(A005117(n)) = A068328(n) = A003415(A005117(n)). - Reinhard Zumkeller, Jul 20 2003

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

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..10000

MathOverflow, A recursion with a number-theoretic function (2019)

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

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

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

CROSSREFS

Cf. A003415, A005117, A068328, A230593, A010051, A000027, A054377.

Sequence in context: A069626 A249274 A205443 * A318320 A014652 A060448

Adjacent sequences:  A069356 A069357 A069358 * A069360 A069361 A069362

KEYWORD

nonn

AUTHOR

Benoit Cloitre, Apr 15 2002

STATUS

approved

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Last modified October 17 08:36 EDT 2019. Contains 328107 sequences. (Running on oeis4.)