A360523 a(n) = Sum_{d|n} mu(rad(d)) * delta_d(n/d), where rad(n) = A007947(n) and delta_d(n) is the greatest divisor of n that is relatively prime to d.

1, 1, 2, 2, 4, 2, 6, 5, 7, 4, 10, 4, 12, 6, 8, 12, 16, 7, 18, 8, 12, 10, 22, 10, 23, 12, 24, 12, 28, 8, 30, 27, 20, 16, 24, 14, 36, 18, 24, 20, 40, 12, 42, 20, 28, 22, 46, 24, 47, 23, 32, 24, 52, 24, 40, 30, 36, 28, 58, 16, 60, 30, 42, 58, 48, 20, 66, 32, 44, 24

Offset

1,3

Comments

Analogous to the Euler totient function (A000010) as A360522 is analogous to A000203.

Links

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

Mizan R. Khan, A variant of the divisor functions sigma_a(n), JP Journal of Algebra, Number Theory and Applications, Vol. 5, No. 3 (2005), pp. 561-574.

Formula

Multiplicative with a(p^e) = p^e - e.

Dirichlet g.f.: zeta(s-1)*zeta(s)^2 * Product_{p prime} (1 - 3/p^s + 1/p^(2*s-1) + 1/p^(2*s)).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - p/((p-1)*(p+1)^2)) = 0.3243742337... .

A000010(n) <= a(n) <= A047994(n) (Khan, 2005).

a(n) = A000010(n) if and only if n is in A078779 (i.e., n is either squarefree or twice a squarefree number).

a(n) = A047994(n) if and only if n is in A005117 (i.e., n is squarefree).

Mathematica

f[p_, e_] := p^e - e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2] - f[i, 2]); }

Crossrefs

Cf. A000010, A000203, A005117, A007947 (rad), A008683 (mu), A047994, A078779, A360522.

Sequence in context: A054929 A236628 A078429 * A358039 A171751 A124676

Adjacent sequences: A360520 A360521 A360522 * A360524 A360525 A360526

Keyword

nonn,mult

Author

Amiram Eldar, Feb 10 2023

Status

approved