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

1,6

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

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)

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 *)

Prog

(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));

Crossrefs

Inverse Möbius transform of A353350.

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

Cf. also A353332, A353354, A353362.

Sequence in context: A049046 A227945 A003647 * A084217 A245548 A025903

Adjacent sequences: A353349 A353350 A353351 * A353353 A353354 A353355

Keyword

nonn

Author

Antti Karttunen, Apr 15 2022

Status

approved