A380449 Sum of the divisors of n plus the number of nonprime divisors of n.

2, 4, 5, 9, 7, 14, 9, 18, 15, 20, 13, 32, 15, 26, 26, 35, 19, 43, 21, 46, 34, 38, 25, 66, 33, 44, 43, 60, 31, 77, 33, 68, 50, 56, 50, 98, 39, 62, 58, 96, 43, 101, 45, 88, 82, 74, 49, 132, 59, 97, 74, 102, 55, 126, 74, 126, 82, 92, 61, 177, 63, 98, 108, 133, 86, 149, 69, 130, 98, 149, 73, 205, 75, 116, 128, 144, 98, 173, 81, 194, 125, 128, 85, 233, 110, 134, 122

Offset: 1

Author

Wesley Ivan Hurt, Jun 21 2025

Keywords

• nonn
• easy

Comments

Inverse Möbius transform of A113636(n).

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

Examples

			The sum of the divisors of 12 = 1+2+3+4+6+12 = 28 and 12 has 4 nonprime divisors, so a(12) = 28 + 4 = 32.
		

Crossrefs

Cf. A000005 (tau), A000203 (sigma), A001221 (omega), A033273, A113636.

Programs

Mathematica

Table[DivisorSigma[0, n] - PrimeNu[n] + DivisorSigma[1, n], {n, 100}]

Formula

a(n) = sigma(n) + tau(n) - omega(n).

a(n) = Sum_{d|n} A113636(d).

a(n) = A000203(n) + A033273(n).

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