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

Author

Amiram Eldar, Feb 10 2023

Keywords

• nonn
• mult

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.

Crossrefs

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

Programs

Mathematica

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

PARI

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

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).