A178638 - OEIS

%I #25 May 28 2017 09:15:46

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

%H Antti Karttunen, <a href="/A178638/b178638.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A000005(n) - A089723(n).

%F 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 For n = 16, set of such divisors is {1, 8}; a(16) = 2.

%t Table[DivisorSum[n, 1 &, If[# > 1, #^IntegerExponent[n, #], 1] != n &], {n, 100}] (* _Michael De Vlieger_, May 27 2017 *)

%o (PARI)

%o A286561(n,k) = if(1==k, 1, valuation(n, k));

%o A178638(n) = sumdiv(n,d,if((d^A286561(n,d))<>n,1,0)); \\ _Antti Karttunen_, May 26 - 27 2017

%o (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 Cf. A000005, A089723, A169594, A186643, A286561.

%K nonn

%O 1,6

%A _Jaroslav Krizek_, Dec 25 2010