login
Number of divisors d of n for which A048675(d) is not a multiple of 3.
5

%I #13 Apr 19 2022 11:41:14

%S 0,1,1,2,1,2,1,2,2,3,1,4,1,2,2,3,1,4,1,4,3,3,1,5,2,2,2,4,1,5,1,4,2,3,

%T 2,6,1,2,3,5,1,5,1,4,4,3,1,6,2,4,2,4,1,5,3,5,3,2,1,8,1,3,4,4,2,5,1,4,

%U 2,5,1,8,1,2,4,4,2,5,1,7,3,3,1,8,3,2,3,5,1,8,3,4,2,3,2,8,1,4,4,6,1,5,1,5,5

%N Number of divisors d of n for which A048675(d) is not a multiple of 3.

%H Antti Karttunen, <a href="/A353351/b353351.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} (1-A353350(d)).

%F a(n) = A000005(n) - A353352(n).

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

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

%F a(n) = A353328(n) + A353329(n).

%t f[p_, e_] := e*2^(PrimePi[p] - 1); q[1] = False; 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 A353351(n) = sumdiv(n,d,!A353350(d));

%Y Cf. A000005, A003961, A048675, A332820, A348717, A353328, A353329, A353350, A353352, A353354.

%Y Cf. also A353361.

%K nonn

%O 1,4

%A _Antti Karttunen_, Apr 15 2022