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A175070 a(n) is the sum of perfect divisors of n - n, where a perfect divisor of n is a divisor d such that d^k = n for some k >= 1.

%I A175070 #13 Dec 12 2024 09:29:17
%S A175070 0,0,0,2,0,0,0,2,3,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,5,0,3,0,0,0,0,2,0,0,
%T A175070 0,6,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,0,0,
%U A175070 0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10
%N A175070 a(n) is the sum of perfect divisors of n - n, where a perfect divisor of n is a divisor d such that d^k = n for some k >= 1.
%C A175070 a(1) = 0, for n >=2: a(n) = sum of perfect divisors of n less than n.
%C A175070 a(n) > 0 for perfect powers n = A001597(m) for m > 2.
%H A175070 Antti Karttunen, <a href="/A175070/b175070.txt">Table of n, a(n) for n = 1..65537</a>
%H A175070 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>
%F A175070 a(n) = A175067(n) - n.
%p A175070 a:= n-> add(`if`(n=d^ilog[d](n), d, 0), d=numtheory[divisors](n) minus {n}):
%p A175070 seq(a(n), n=1..100);  # _Alois P. Heinz_, Dec 12 2024
%o A175070 (PARI) A175070(n) = if(!ispower(n),0,sumdiv(n,d,if((d>1)&&(d<n)&&((d^valuation(n,d))==n),d,0))); \\ _Antti Karttunen_, Jun 12 2018
%o A175070 (PARI) first(n) = {my(res = vector(n)); for(i = 2, sqrtint(n), for(j = 2, logint(n, i), res[i^j] += i)); res} \\ _David A. Corneth_, Jun 12 2018
%Y A175070 Cf. A175067.
%K A175070 nonn
%O A175070 1,4
%A A175070 _Jaroslav Krizek_, Jan 23 2010