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A284521 Sum of largest prime power factors of numbers <= n.

%I A284521 #9 Mar 29 2017 19:46:09
%S A284521 1,3,6,10,15,18,25,33,42,47,58,62,75,82,87,103,120,129,148,153,160,
%T A284521 171,194,202,227,240,267,274,303,308,339,371,382,399,406,415,452,471,
%U A284521 484,492,533,540,583,594,603,626,673,689,738,763,780,793,846,873,884,892,911,940,999,1004,1065,1096,1105,1169,1182
%N A284521 Sum of largest prime power factors of numbers <= n.
%C A284521 Partial sums of A034699.
%H A284521 Robert Israel, <a href="/A284521/b284521.txt">Table of n, a(n) for n = 1..10000</a>
%F A284521 Conjecture: a(n) = O(n^2/log(n)).
%e A284521 a(1) = 1;
%e A284521 a(2) = 3 because 2 is a prime and 1 + 2 = 3;
%e A284521 a(3) = 6 because 3 is a prime and 3 + 3 = 6;
%e A284521 a(4) = 10 because 4 = 2^2 and 6 + 4 = 10;
%e A284521 a(5) = 15 because 5 is a prime and 10 + 5 = 15;
%e A284521 a(6) = 18 because 12 = 2*3 and 15 + 3 = 18, etc.
%p A284521 g:= n -> max(map(t -> t[1]^t[2], ifactors(n)[2])): g(1):= 1:
%p A284521 ListTools:-PartialSums(map(g, [$1..100])); # _Robert Israel_, Mar 29 2017
%t A284521 Accumulate[Join[{1}, Table[Last[Select[Divisors[n], PrimePowerQ[#1] & ]], {n, 2, 65}]]]
%o A284521 (PARI) a(n) = if (n==1, 1, 1+ sum(k=2, n, f = factor(k); f[#f~,1]^f[#f~,2])); \\ _Michel Marcus_, Mar 28 2017
%Y A284521 Cf. A034699, A046670, A073355.
%K A284521 nonn
%O A284521 1,2
%A A284521 _Ilya Gutkovskiy_, Mar 28 2017