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A353352 Number of divisors d of n for which A048675(d) is a multiple of 3.

%I A353352 #20 Apr 28 2022 07:47:36
%S A353352 1,1,1,1,1,2,1,2,1,1,1,2,1,2,2,2,1,2,1,2,1,1,1,3,1,2,2,2,1,3,1,2,2,1,
%T A353352 2,3,1,2,1,3,1,3,1,2,2,1,1,4,1,2,2,2,1,3,1,3,1,2,1,4,1,1,2,3,2,3,1,2,
%U A353352 2,3,1,4,1,2,2,2,2,3,1,3,2,1,1,4,1,2,1,3,1,4,1,2,2,1,2,4,1,2,2,3,1,3,1,3,3
%N A353352 Number of divisors d of n for which A048675(d) is a multiple of 3.
%C A353352 a(n) is the number of terms of A332820 that divide n.
%H A353352 Antti Karttunen, <a href="/A353352/b353352.txt">Table of n, a(n) for n = 1..65537</a>
%H A353352 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A353352 a(n) = Sum_{d|n} A353350(d).
%F A353352 a(n) = A000005(n) - A353351(n).
%F A353352 a(p) = 1 for all primes p.
%F A353352 a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.
%F A353352 From _Peter Munn_, Apr 22 2022: (Start)
%F A353352 a(n) = A353328(n) = A353329(n) iff 3|A000005(n) [i.e., A353470(n) = 1].
%F A353352 Otherwise a(n) = A353328(n) iff A048675(n) == 1 (mod 3); a(n) = A353329(n) iff A048675(n) == 2 (mod 3).
%F A353352 (End)
%t A353352 f[p_, e_] := e*2^(PrimePi[p] - 1); q[1] = True; q[n_] := Divisible[Plus @@ f @@@ FactorInteger[n], 3]; a[n_] := DivisorSum[n, 1 &, q[#] &]; Array[a, 100] (* _Amiram Eldar_, Apr 15 2022 *)
%o A353352 (PARI)
%o A353352 A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
%o A353352 A353350(n) = (0==(A048675(n)%3));
%o A353352 A353352(n) = sumdiv(n,d,A353350(d));
%Y A353352 Inverse Möbius transform of A353350.
%Y A353352 Cf. A000005, A003961, A048675, A059269, A332820, A348717, A353328, A353329, A353351, A353470.
%Y A353352 Cf. also A353332, A353354, A353362.
%K A353352 nonn
%O A353352 1,6
%A A353352 _Antti Karttunen_, Apr 15 2022