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A008472
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Sum of the distinct primes dividing n.
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197
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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
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OFFSET
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1,2
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COMMENTS
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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 [From Gary W. Adamson, Aug 23 2008]
a(n) = n iff n is prime. [From Daniel Forgues, Mar 24 2009]
a(n) = n is a new record iff n is prime. [From 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]
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LINKS
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Franklin T. Adams-Watters and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from Franklin T. Adams-Watters)
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FORMULA
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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))). [From Franklin T. Adams-Watters, Sep 01 2009]
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MAPLE
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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
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MATHEMATICA
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Prepend[ Array[ Plus @@ First[ Transpose[ FactorInteger[ # ] ] ]&, 100, 2 ], 0 ]
Join[{0}, Rest[Total[Transpose[FactorInteger[#]][[1]]]&/@Range[100]]] (* Harvey P. Dale, Jun 18 2012 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,nice
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AUTHOR
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Olivier Gérard
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STATUS
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approved
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