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A230593
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a(n) = n * Sum_{q|n} 1 / q, where q are noncomposite numbers (A008578) dividing n.
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6
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1, 3, 4, 6, 6, 11, 8, 12, 12, 17, 12, 22, 14, 23, 23, 24, 18, 33, 20, 34, 31, 35, 24, 44, 30, 41, 36, 46, 30, 61, 32, 48, 47, 53, 47, 66, 38, 59, 55, 68, 42, 83, 44, 70, 69, 71, 48, 88, 56, 85, 71, 82, 54, 99, 71, 92, 79, 89, 60, 122, 62, 95, 93, 96, 83, 127
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OFFSET
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1,2
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LINKS
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FORMULA
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For n > 1, a(n) = n + n * Sum_(p|n) 1 / p, where p are primes dividing n.
a(n) = A080339(n) * A000027(n), where operation * denotes Dirichlet convolution, i.e. convolution of type: a(n) = Sum_{d|n} b(d) * c(n/d).
For p, q = distinct primes, a(p) = p + 1, a(pq) = pq - 1.
(End)
For p prime, k>=1, a(p^k) = p^(k-1) * (p+1). - Bernard Schott, Nov 12 2021
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EXAMPLE
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For n = 6: a(6) = 6 * (1/1 + 1/2 + 1/3) = 11.
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MATHEMATICA
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a[n_] := n * DivisorSum[n, 1/# &, !CompositeQ[#] &]; Array[a, 100] (* Amiram Eldar, Nov 12 2021 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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