A318499 -id:A318499 - cp's OEIS frontend

A318498 Denominators of the sequence whose Dirichlet convolution with itself yields A061389, number of (1+phi)-divisors of n.

1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 8, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 8, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 16, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 8, 8, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 8, 1, 2, 2, 4, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Aug 30 2018

The sequence seems to give the denominators of a few other similarly constructed rational valued sequences obtained as "Dirichlet Square Roots" (of possibly A092520 and A293443).

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A061389, A318497 (numerators), A318499.

Cf. also A299150, A046644.

up_to = 65537;
A061389(n) = factorback(apply(e -> (1+eulerphi(e)),factor(n)[,2]));
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA317937.
v318497_98 = DirSqrt(vector(up_to, n, A061389(n)));
A318497(n) = numerator(v318497_98[n]);
A318498(n) = denominator(v318497_98[n]);

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A061389(n) - Sum_{d|n, d>1, d 1.

a(n) = 2^A318499(n).

A372503 The number of prime powers that are noninfinitary divisors of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 3, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 3, 3, 0, 0, 1, 0, 0, 0

Offset: 1

Amiram Eldar, May 04 2024

First differs from A318499 at n = 32.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000120, A001222, A036537, A048967, A162643, A318499, A349258.

f[p_, e_] :=  e + 1 - 2^DigitCount[e, 2, 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecsum(apply(x -> x + 1 - 1 << hammingweight(x), factor(n)[, 2]));

Additive with a(p^e) = e - 2^A000120(e) + 1 = A048967(e).
a(n) = A001222(n) - A349258(n).
a(n) = 0 if and only if n is in A036537.
a(n) > 0 if and only if n is in A162643.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} f(1/p) = 0.4917971717413486467..., where f(x) = 1/(1-x) - (1-x) * Product_{k>=0} (1 + 2*x^(2^k)).

cp's OEIS Frontend

A318498 Denominators of the sequence whose Dirichlet convolution with itself yields A061389, number of (1+phi)-divisors of n.

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