A348155 a(n) = tau(n)^2 + omega(n)*(sigma(n) - tau(n)).

1, 5, 6, 13, 8, 32, 10, 27, 19, 44, 14, 80, 16, 56, 56, 51, 20, 102, 22, 108, 72, 80, 26, 168, 37, 92, 52, 136, 32, 256, 34, 93, 104, 116, 104, 245, 40, 128, 120, 228, 44, 328, 46, 192, 180, 152, 50, 328, 63, 210, 152, 220, 56, 288, 152, 288, 168, 188, 62, 612, 64, 200, 232, 169

Offset

1,2

Comments

For each ordered pair of divisors of n, (d1,d2), a(n) can also be found using the algorithm: add d1 if d2 is prime; otherwise add 1. For example, when n = 4 the divisor pairs are: (1,1), (1,2), (1,4), (2,1), (2,2), (2,4), (4,1), (4,2), (4,4) which gives 1 + 1 + 1 + 1 + 2 + 1 + 1 + 4 + 1 = 13.

Links

Table of n, a(n) for n=1..64.

Formula

a(n) = Sum_{d1|n} Sum_{d2|n} d1^c(d2), where c = A010051.

a(prime(n)) = prime(n) + 3.

Mathematica

Table[DivisorSigma[0, n] (DivisorSigma[0, n] - PrimeNu[n]) + PrimeNu[n] DivisorSigma[1, n], {n, 80}]

Prog

(PARI) a(n) = my(f=factor(n), d=numdiv(f)); d^2 + omega(f)*(sigma(f) - d); \\ Michel Marcus, Oct 05 2021

Crossrefs

Cf. A000005 (tau), A000040 (n-th prime), A000203 (sigma), A001221 (omega), A010051 (prime characteristic).

Sequence in context: A099473 A051572 A344221 * A047436 A334320 A061437

Adjacent sequences: A348152 A348153 A348154 * A348156 A348157 A348158

Keyword

nonn

Author

Wesley Ivan Hurt, Oct 03 2021

Status

approved