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A178638 a(n) is the number of divisors d of n such that d^k is not equal to n for any k >= 1.

%I A178638 #25 May 28 2017 09:15:46
%S A178638 0,1,1,1,1,3,1,2,1,3,1,5,1,3,3,2,1,5,1,5,3,3,1,7,1,3,2,5,1,7,1,4,3,3,
%T A178638 3,7,1,3,3,7,1,7,1,5,5,3,1,9,1,5,3,5,1,7,3,7,3,3,1,11,1,3,5,3,3,7,1,5,
%U A178638 3,7,1,11,1,3,5,5,3,7,1,9,2,3,1,11,3,3,3,7,1,11,3,5,3,3,3,11,1,5,5,7
%N A178638 a(n) is the number of divisors d of n such that d^k is not equal to n for any k >= 1.
%H A178638 Antti Karttunen, <a href="/A178638/b178638.txt">Table of n, a(n) for n = 1..10000</a>
%F A178638 a(n) = A000005(n) - A089723(n).
%F A178638 a(1) = 0, a(p) = 1, a(pq) = 3, a(pq...z) = 2^k-1, a(p^k) = k+1-A000005(k), for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.
%e A178638 For n = 16, set of such divisors is {1, 8}; a(16) = 2.
%t A178638 Table[DivisorSum[n, 1 &, If[# > 1, #^IntegerExponent[n, #], 1] != n &], {n, 100}] (* _Michael De Vlieger_, May 27 2017 *)
%o A178638 (PARI)
%o A178638 A286561(n,k) = if(1==k, 1, valuation(n, k));
%o A178638 A178638(n) = sumdiv(n,d,if((d^A286561(n,d))<>n,1,0)); \\ _Antti Karttunen_, May 26 - 27 2017
%o A178638 (PARI) a(n) = if(n==1, return(0)); my(f=factor(n), g = f[1, 2]); for(i=2, matsize(f)[1], g=gcd(g, f[i, 2])); numdiv(n) - numdiv(g) \\ _David A. Corneth_, May 27 2017
%Y A178638 Cf. A000005, A089723, A169594, A186643, A286561.
%K A178638 nonn
%O A178638 1,6
%A A178638 _Jaroslav Krizek_, Dec 25 2010