A380447 Number of primes dividing n plus n times the number of nonprimes dividing n.

1, 3, 4, 9, 6, 14, 8, 25, 19, 22, 12, 50, 14, 30, 32, 65, 18, 74, 20, 82, 44, 46, 24, 146, 51, 54, 82, 114, 30, 153, 32, 161, 68, 70, 72, 254, 38, 78, 80, 242, 42, 213, 44, 178, 182, 94, 48, 386, 99, 202, 104, 210, 54, 326, 112, 338, 116, 118, 60, 543, 62, 126, 254, 385, 132, 333, 68, 274, 140, 353, 72, 722, 74, 150, 302, 306, 156, 393, 80, 642, 325, 166

Offset

1,2

Comments

For each divisor d of n, add 1 if d is prime, else add n.

Inverse Möbius transform of A385313(n).

Links

Table of n, a(n) for n=1..82.

Formula

a(n) = n*tau(n) - (n-1)*omega(n).

a(n) = Sum_{d|n} n^c(d), where c = A005171.

a(n) = Sum_{d|n} A385313(d).

a(p^k) = 1 + k*p^k for p prime and k>=1. - Wesley Ivan Hurt, Jul 02 2025

a(n) = A001221(n) - n*A033273(n). - Wesley Ivan Hurt, Aug 31 2025

Example

12 has 2 prime divisors and 4 nonprime divisors, so a(12) = (2) + 12*(4) = 50.

Mathematica

Table[n*DivisorSigma[0, n] - (n - 1) PrimeNu[n], {n, 100}]

Crossrefs

Cf. A000005 (tau), A001221 (omega), A005171, A033273.

Sequence in context: A337180 A003959 A344461 * A168341 A083111 A354112

Adjacent sequences: A380444 A380445 A380446 * A380448 A380449 A380450

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jun 21 2025

Status

approved