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%I #10 Oct 26 2021 16:52:53
%S 1,1,1,1,1,3,1,2,1,3,1,4,1,3,3,2,1,4,1,4,3,3,1,6,1,3,2,4,1,7,1,4,3,3,
%T 3,5,1,3,3,6,1,7,1,4,4,3,1,7,1,4,3,4,1,6,3,6,3,3,1,10,1,3,4,3,3,7,1,4,
%U 3,7,1,8,1,3,4,4,3,7,1,7,2,3,1,10,3,3,3,6,1,10,3,4,3,3,3,10,1,4,4,5
%N Number of non-exponential divisors of n.
%C The non-exponential divisors d|n of a number n= p(i)^e(i) are divisors d not of the form p(i)^s(i), s(i)|e(i) for all i.
%H Antti Karttunen, <a href="/A160097/b160097.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = A000005(n) - A049419(n) for n >= 2.
%F a(1) = 1, a(p) = 1, a(p*q) = 3, a(p*q*...*z) = 2^k - 1, where the indices are p=primes (A000040), p*q = product of two distinct primes (A006881), and generally p*q*...*z = product of k (k > 0) distinct primes (A120944).
%F a(p^k) = k + 1 - A000005(k), where p are primes (A000040), p^k are prime powers A000961(n>1), k = natural numbers (A000027).
%F a(p^q) = q - 1, where p and q are primes (A000040), and p^q = prime powers of primes (A053810).
%e a(8)=2 because 1 and 2^2 are non-exponential divisors of 8=2^3. 2^2 is a non-exponential divisor because 2^2=4 divides 8, but the exponent 2=s(1) does not divide the exponent 3=e(1).
%t f1[p_, e_] := e + 1; f2[p_, e_] := DivisorSigma[0, e]; a[1] = 1; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; Array[a, 100] (* _Amiram Eldar_, Oct 26 2021 *)
%o (PARI)
%o A049419(n) = { my(f = factor(n), m = 1); for(k=1, #f~, m *= numdiv(f[k, 2])); m; } \\ After Jovovic's formula for A049419.
%o A160097(n) = if(1==n,n,(numdiv(n) - A049419(n))); \\ _Antti Karttunen_, May 25 2017
%Y Cf. A000005, A049419, A000040, A006881, A120944, A000961, A053810.
%K nonn
%O 1,6
%A _Jaroslav Krizek_, May 01 2009
%E Edited by _R. J. Mathar_, May 08 2009
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