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A008472 Sum of the distinct primes dividing n. 226
0, 2, 3, 2, 5, 5, 7, 2, 3, 7, 11, 5, 13, 9, 8, 2, 17, 5, 19, 7, 10, 13, 23, 5, 5, 15, 3, 9, 29, 10, 31, 2, 14, 19, 12, 5, 37, 21, 16, 7, 41, 12, 43, 13, 8, 25, 47, 5, 7, 7, 20, 15, 53, 5, 16, 9, 22, 31, 59, 10, 61, 33, 10, 2, 18, 16, 67, 19, 26, 14, 71, 5, 73 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Sometimes called sopf(n).

Sum of primes dividing n (without repetition) (compare A001414).

Equals A051731 * A061397 = inverse Mobius transform of [0, 2, 3, 0, 5, 0, 7,...]. - Gary W. Adamson, Feb 14 2008

Equals row sums of triangle A143535. - Gary W. Adamson, Aug 23 2008

a(n) = n if and only if n is prime. - Daniel Forgues, Mar 24 2009

a(n) = n is a new record if and only if n is prime. - Zak Seidov, Jun 27 2009

a(A001043(n)) = A191583(n);

For n > 0: a(A000079(n)) = 2, a(A000244(n)) = 3, a(A000351(n)) = 5, a(A000420(n)) = 7;

a(A006899(n)) <= 3; a(A003586(n)) = 5; a(A033846(n)) = 7; a(A033849(n)) = 8; a(A033847(n)) = 9; a(A033850(n)) = 10; a(A143207(n)) = 10. - Reinhard Zumkeller, Jun 28 2011

For n > 1: a(n) = Sum(A027748(n,k): 1 <= k <= A001221(n)). - Reinhard Zumkeller, Aug 27 2011

If n is the product of twin primes (A037074), a(n) = 2*sqrt(n+1) = sqrt(4n+4). - Wesley Ivan Hurt, Sep 07 2013

LINKS

Franklin T. Adams-Watters and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from Franklin T. Adams-Watters)

FORMULA

n = Product(p_j^k_j) -> Sum (p_j).

Additive with a(p^e) = p.

G.f.: sum(k >= 1, prime(k)*x^prime(k)/(1-x^prime(k))). - Franklin T. Adams-Watters, Sep 01 2009

EXAMPLE

a(18) = 5 because 18 = 2 * 3^2 and 2 + 3 = 5.

a(19) = 19 because 19 is prime.

a(20) = 7 because 20 = 2^2 * 5 and 2 + 5 = 7.

MAPLE

A008472 := n -> add(d, d = select(isprime, numtheory[divisors](n))):

seq(A008472(i), i = 1..40); # Peter Luschny, Jan 31 2012

A008472 := proc(n)

        add( d, d= numtheory[factorset](n)) ;

end proc: # R. J. Mathar, Jul 08 2012

MATHEMATICA

Prepend[Array[Plus @@ First[Transpose[FactorInteger[#]]] &, 100, 2], 0]

Join[{0}, Rest[Total[Transpose[FactorInteger[#]][[1]]]&/@Range[100]]] (* Harvey P. Dale, Jun 18 2012 *)

(* Requires version 7.0+ *) Table[DivisorSum[n, # &, PrimeQ[#] &], {n, 75}] (* Alonso del Arte, Dec 13 2014 *)

PROG

(PARI) sopf(n) = local(fac, i); fac=factor(n); sum(i=1, matsize(fac)[1], fac[i, 1])

(Sage)

def A008472(n) :

    D = filter(is_prime, divisors(n))

    return add(d for d in D)

print [A008472(i) for i in (1..40)] # Peter Luschny, Jan 31 2012

(Haskell)

a008472 = sum . a027748_row  -- Reinhard Zumkeller, Mar 29 2012

CROSSREFS

Cf. A001414 (sopfr), A001222, A051731, A061397, A143535, A085020.

Sequence in context: A095402 A086294 A075860 * A123528 A074036 A074251

Adjacent sequences:  A008469 A008470 A008471 * A008473 A008474 A008475

KEYWORD

nonn,nice,changed

AUTHOR

Olivier Gérard

STATUS

approved

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Last modified December 22 11:41 EST 2014. Contains 252355 sequences.