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A175067 a(n) is the sum of perfect divisors of n, where a perfect divisor of n is a divisor d such that d^k = n for some k >= 1.

%I A175067 #20 Dec 12 2024 15:38:40
%S A175067 1,2,3,6,5,6,7,10,12,10,11,12,13,14,15,22,17,18,19,20,21,22,23,24,30,
%T A175067 26,30,28,29,30,31,34,33,34,35,42,37,38,39,40,41,42,43,44,45,46,47,48,
%U A175067 56,50,51,52,53,54,55,56,57,58,59,60,61,62,63,78,65,66,67,68,69,70,71,72
%N A175067 a(n) is the sum of perfect divisors of n, where a perfect divisor of n is a divisor d such that d^k = n for some k >= 1.
%C A175067 a(n) > n for perfect powers n = A001597(m) for m > 2.
%H A175067 Ivan Neretin, <a href="/A175067/b175067.txt">Table of n, a(n) for n = 1..10000</a>
%F A175067 a(n) = A175070(n) + n. [_Jaroslav Krizek_, Jan 24 2010]
%F A175067 From _Ridouane Oudra_, Dec 12 2024: (Start)
%F A175067 a(n) = n, for n in A007916.
%F A175067 a(n^m) = Sum_{d|m} n^d, for n in A007916 and m an integer >0.
%F A175067 More generally, for all integers n we have :
%F A175067 a(n) = Sum_{d|A253641(n)} n^(d/A253641(n)).
%F A175067 a(n) = Sum_{d|A253641(n)} A052410(n)^d. (End)
%e A175067 For n = 8: a(8) = 10; there are two perfect divisors of 8: 2 and 8; their sum is 10.
%p A175067 A175067:= proc(n) local a, d, k; if n=1 then return 1 end if; a:=0; for d in numtheory[divisors](n) minus {1} do for k do if d^k=n then a:= a+d end if; if n <= d^k then break; end if; end do; end do; a end proc:
%p A175067 seq(A175067(n), n=1..80); # _Ridouane Oudra_, Dec 12 2024
%p A175067 # second Maple program:
%p A175067 a:= n-> add(`if`(n=d^ilog[d](n), d, 0), d=numtheory[divisors](n)):
%p A175067 seq(a(n), n=1..72);  # _Alois P. Heinz_, Dec 12 2024
%t A175067 Table[Plus @@ (n^(1/Divisors[GCD @@ FactorInteger[n][[All, 2]]])), {n, 72}] (* _Ivan Neretin_, May 13 2015 *)
%Y A175067 Cf. A175070, A007916, A175068, A253641, A052410.
%K A175067 nonn
%O A175067 1,2
%A A175067 _Jaroslav Krizek_, Jan 23 2010
%E A175067 Name edited by _Michel Marcus_, Jun 13 2018