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

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, 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, 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

Offset: 1

Author

Antti Karttunen, Apr 15 2022

Keywords

• nonn

Comments

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

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Inverse Möbius transform of A353350.

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

Cf. also A353332, A353354, A353362.

Programs

Mathematica

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 *)

PARI

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A353350(n) = (0==(A048675(n)%3));
A353352(n) = sumdiv(n,d,A353350(d));

Formula

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

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

a(p) = 1 for all primes p.

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

From Peter Munn, Apr 22 2022: (Start)

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

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

(End)