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%I #30 Sep 08 2022 08:46:23
%S 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,
%T 9,29,29,40,31,10,28,38,24,30,37,42,32,32,41,48,43,41,27,50,47,31,14,
%U 26,40,47,53,26,32,40,44,62,59,64,61,66,33,12,36,64,67,59,52,56
%N a(n) = Sum_{d|n} Sum_{p|d, p prime} p.
%C Inverse Möbius transform of A008472.
%H David A. Corneth, <a href="/A319131/b319131.txt">Table of n, a(n) for n = 1..10000</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F G.f.: Sum_{k>=1} A008472(k)*x^k/(1 - x^k).
%F L.g.f.: -log(Product_{k>=1} (1 - x^k)^(A008472(k)/k)) = Sum_{n>=1} a(n)*x^n/n.
%F a(p^k) = p*k, where p is a prime.
%F a(n) = Sum_{p|n} p*tau(n/p), where p is a prime and tau(n) = A000005(n). - _Ridouane Oudra_, Oct 08 2019
%F 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
%F a(n) = Sum_{d|n} sopf(d). - _Wesley Ivan Hurt_, May 23 2021
%e a(12) = 13 as 12 has 6 divisors and 2 * 6 * (2/3) + 3 * 6 * (1/2) = 17. - _David A. Corneth_, Oct 08 2019
%p with(numtheory): seq(add(p*tau(n/p), p in factorset(n)), n=1..80); # _Ridouane Oudra_, Oct 08 2019
%t Table[Sum[Total[Select[Divisors[d], PrimeQ]], {d, Divisors[n]}], {n, 70}]
%t nmax = 70; Rest[CoefficientList[Series[Sum[DivisorSum[k, # &, PrimeQ[#] &] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
%t nmax = 70; Rest[CoefficientList[Series[-Log[Product[(1 - x^k)^(DivisorSum[k, # &, PrimeQ[#] &]/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
%o (PARI) a(n) = sumdiv(n, d, my(f=factor(d)); vecsum(f[,1])); \\ _Michel Marcus_, Oct 08 2019
%o (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
%o (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
%o (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)
%Y Cf. A000005, A007429, A008472, A061397, A062799, A319132.
%K nonn,easy
%O 1,2
%A _Ilya Gutkovskiy_, Sep 11 2018
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