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A324179 Number of distinct values A297167 obtains over divisors > 1 of n, minus number of prime factors of n counted with multiplicity: a(n) = A324190(n) - A001222(n).

%I A324179 #13 Feb 19 2019 18:19:43
%S A324179 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,
%T A324179 0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,1,
%U A324179 0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,2,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,2,0
%N A324179 Number of distinct values A297167 obtains over divisors > 1 of n, minus number of prime factors of n counted with multiplicity: a(n) = A324190(n) - A001222(n).
%C A324179 a(n) is zero for all prime powers (A000961), but also for many other numbers.
%H A324179 Antti Karttunen, <a href="/A324179/b324179.txt">Table of n, a(n) for n = 1..65537</a>
%H A324179 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A324179 a(n) = A324190(n) - A001222(n).
%F A324179 a(n) <= A324192(n).
%e A324179 Divisors of 56 larger than 1 are [2, 4, 7, 8, 14, 28, 56]. When A297167 is applied to each, one obtains values: [0, 1, 3, 2, 3, 4, 5], of which 6 values are distinct (as one of them, 3, occurs twice). On the other hand, 56 = 2 * 2 * 2 * 7 has four prime factors in total, thus a(56) = 6 - 4 = 2.
%o A324179 (PARI)
%o A324179 A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
%o A324179 A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
%o A324179 A324190(n) = #Set(apply(A297167, select(d -> d>1,divisors(n))));
%o A324179 A324179(n) = (A324190(n)-bigomega(n));
%Y A324179 Cf. A000961, A001222, A061395, A297112, A297167, A324120, A324190, A324191, A324192.
%K A324179 nonn
%O A324179 1,56
%A A324179 _Antti Karttunen_, Feb 19 2019