A248577 Product of the number of divisors of n and the number of distinct prime divisors of n; i.e., tau(n) * omega(n).
0, 2, 2, 3, 2, 8, 2, 4, 3, 8, 2, 12, 2, 8, 8, 5, 2, 12, 2, 12, 8, 8, 2, 16, 3, 8, 4, 12, 2, 24, 2, 6, 8, 8, 8, 18, 2, 8, 8, 16, 2, 24, 2, 12, 12, 8, 2, 20, 3, 12, 8, 12, 2, 16, 8, 16, 8, 8, 2, 36, 2, 8, 12, 7, 8, 24, 2, 12, 8, 24, 2, 24, 2, 8, 12, 12, 8, 24
Offset: 1
Examples
a(6) = 8; 6 has four divisors {1,2,3,6} and two distinct prime divisors {2,3}, so a(6) = 4*2 = 8.
a(9) = 3; 9 has three divisors {1,3,9} and 1 distinct prime divisor {3}, so a(9) = 3*1 = 3.
a(12) = 12; 12 has 6 divisors {1,2,3,4,6,12} and 2 distinct prime divisors {2,3}, so a(12) = 6*2 = 12.
Links
- G. C. Greubel, Table of n, a(n) for n = 1..5000
- Tanay Wakhare, Sums involving the number of distinct prime factors function, arXiv:1604.05671 [math.HO], 2016-2017.
Programs
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Maple
with(numtheory): A248577:=n->tau(n)*nops(factorset(n)): seq(A248577(n), n=1..100);
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Mathematica
Table[DivisorSigma[0, n] PrimeNu[n], {n, 100}] -
PARI
vector(100, n, numdiv(n)*omega(n)) \\ Michel Marcus, Oct 09 2014
Formula
If n is squarefree, then a(n) = omega(n)*2^omega(n). - Wesley Ivan Hurt, Jun 09 2020
Dirichlet g.f.: zeta(s)^2 * (2*P(s) - P(2*s)), where P(s) is the prime zeta function (Wakhare, 2016). - Amiram Eldar, Sep 19 2023