%I #11 May 20 2017 21:51:57
%S 1,3,4,9,6,12,8,19,16,18,12,30,14,24,24,41,18,42,20,44,32,36,24,64,36,
%T 42,46,58,30,72,32,75,48,54,48,102,38,60,56,94,42,96,44,86,81,72,48,
%U 134,64,98,72,100,54,126,72,124,80,90,60,170,62,96,107,153,84,144,68,128,96
%N Sum of divisors of n weighted by divisor multiplicity in n.
%C If d > 1 divides n, the multiplicity of d in n is the largest integer i such that d^i divides n; e.g. the multiplicity of 4 in 16 is 2. If d = 1 (degenerate case), then the multiplicity of d is defined as 1.
%H Ivan Neretin, <a href="/A168512/b168512.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = Sum_{d|n} A286561(n,d)*d. - _Antti Karttunen_, May 20 2017
%e The divisors of 16 are 1, 2, 4, 8, 16, which are of multiplicity 1, 4, 2, 1, 1, respectively, in 16. So a(16) = 1*1 + 4*2 + 2*4 + 1*8 + 1*16 = 41.
%t Table[1 + Total[Function[i, i*Select[Range[Log[i, n]], Divisible[n, i^#] &][[-1]]] /@ Rest@Divisors@n], {n, 69}] (* _Ivan Neretin_, Jul 26 2015 *)
%t Table[1 + DivisorSum[n, # IntegerExponent[n, #] &, # > 1 &], {n, 69}] (* _Michael De Vlieger_, May 20 2017 *)
%o (PARI)
%o A286561(n,k) = { my(i=1); if(1==k, 1, while(!(n%(k^i)), i = i+1); (i-1)); };
%o A168512(n) = sumdiv(n,d,A286561(n,d)*d); \\ _Antti Karttunen_, May 20 2017
%Y Cf. A169594, A286561, A286563.
%K nonn
%O 1,2
%A _Joseph L. Pe_, Nov 28 2009
%E Extended by _Ray Chandler_, Dec 08 2009