A353352 - OEIS

%I #20 Apr 28 2022 07:47:36

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

%C a(n) is the number of terms of A332820 that divide n.

%H Antti Karttunen, <a href="/A353352/b353352.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%F a(n) = Sum_{d|n} A353350(d).

%F a(n) = A000005(n) - A353351(n).

%F a(p) = 1 for all primes p.

%F a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.

%F From _Peter Munn_, Apr 22 2022: (Start)

%F a(n) = A353328(n) = A353329(n) iff 3|A000005(n) [i.e., A353470(n) = 1].

%F Otherwise a(n) = A353328(n) iff A048675(n) == 1 (mod 3); a(n) = A353329(n) iff A048675(n) == 2 (mod 3).

%F (End)

%t 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 (PARI)

%o A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };

%o A353350(n) = (0==(A048675(n)%3));

%o A353352(n) = sumdiv(n,d,A353350(d));

%Y Inverse Möbius transform of A353350.

%Y Cf. A000005, A003961, A048675, A059269, A332820, A348717, A353328, A353329, A353351, A353470.

%Y Cf. also A353332, A353354, A353362.

%K nonn

%O 1,6

%A _Antti Karttunen_, Apr 15 2022