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A319131
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a(n) = Sum_{d|n} Sum_{p|d, p prime} p.
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7
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0, 2, 3, 4, 5, 10, 7, 6, 6, 14, 11, 17, 13, 18, 16, 8, 17, 18, 19, 23, 20, 26, 23, 24, 10, 30, 9, 29, 29, 40, 31, 10, 28, 38, 24, 30, 37, 42, 32, 32, 41, 48, 43, 41, 27, 50, 47, 31, 14, 26, 40, 47, 53, 26, 32, 40, 44, 62, 59, 64, 61, 66, 33, 12, 36, 64, 67, 59, 52, 56
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OFFSET
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1,2
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COMMENTS
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Inverse Möbius transform of A008472.
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LINKS
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FORMULA
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G.f.: Sum_{k>=1} A008472(k)*x^k/(1 - x^k).
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(A008472(k)/k)) = Sum_{n>=1} a(n)*x^n/n.
a(p^k) = p*k, where p is a prime.
a(n) = Sum_{p|n} p*tau(n)*(e_p-1)/(e_p) where e_p is the exponent of p in the factorization of n. - David A. Corneth, Oct 08 2019
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EXAMPLE
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a(12) = 13 as 12 has 6 divisors and 2 * 6 * (2/3) + 3 * 6 * (1/2) = 17. - David A. Corneth, Oct 08 2019
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MAPLE
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with(numtheory): seq(add(p*tau(n/p), p in factorset(n)), n=1..80); # Ridouane Oudra, Oct 08 2019
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MATHEMATICA
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Table[Sum[Total[Select[Divisors[d], PrimeQ]], {d, Divisors[n]}], {n, 70}]
nmax = 70; Rest[CoefficientList[Series[Sum[DivisorSum[k, # &, PrimeQ[#] &] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 70; Rest[CoefficientList[Series[-Log[Product[(1 - x^k)^(DivisorSum[k, # &, PrimeQ[#] &]/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
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PROG
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(PARI) a(n) = sumdiv(n, d, my(f=factor(d)); vecsum(f[, 1])); \\ Michel Marcus, Oct 08 2019
(PARI) a(n) = my(f = factor(n), nd = numdiv(f)); sum(i = 1, #f~, f[i, 1] * nd / (f[i, 2] + 1) * f[i, 2]) \\ David A. Corneth, Oct 08 2019
(Magma) [0] cat [&+[&+[PrimeDivisors(d)[i]:i in [1..#PrimeDivisors(d)]]:d in Set(Divisors(n)) diff {1}]:n in [2..70]]; // Marius A. Burtea, Oct 08 2019
(Magma) [0] cat [&+[p*#Divisors(n div p):p in PrimeDivisors(n)]:n in [2..70]]; // Marius A. Burtea, Oct 08 2019 (According to the formula given by Ridouane Oudra)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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