A328260 a(n) = n * omega(n).

0, 2, 3, 4, 5, 12, 7, 8, 9, 20, 11, 24, 13, 28, 30, 16, 17, 36, 19, 40, 42, 44, 23, 48, 25, 52, 27, 56, 29, 90, 31, 32, 66, 68, 70, 72, 37, 76, 78, 80, 41, 126, 43, 88, 90, 92, 47, 96, 49, 100, 102, 104, 53, 108, 110, 112, 114, 116, 59, 180, 61, 124, 126, 64, 130, 198, 67, 136, 138, 210

Offset: 1

Author

Ilya Gutkovskiy, Oct 09 2019

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• Mikhail R. Gabdullin and Vitalii V. Iudelevich, Numbers of the form kf(k), arXiv:2201.09287 [math.NT] (2022).
• Eric Weisstein's World of Mathematics, Distinct Prime Factors

Crossrefs

Cf. A001221, A001222, A007947, A038040, A061397, A066959, A069359, A246655 (fixed points), A293548, A298473.

Programs

Magma

[0] cat [n*(#PrimeDivisors(n)):n in [2..70]]; // Marius A. Burtea, Oct 10 2019

Mathematica

Table[n PrimeNu[n], {n, 1, 70}]
nmax = 70; CoefficientList[Series[Sum[Prime[k] x^Prime[k]/(1 - x^Prime[k])^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

PARI

a(n)=n*omega(n) \\ Charles R Greathouse IV, Mar 16 2022

Formula

G.f.: Sum_{k>=1} prime(k) * x^prime(k) / (1 - x^prime(k))^2.

a(n) = bigomega(rad(n)^n).

a(n) = Sum_{d|n} A061397(n/d) * d.

Define f(x) = #{n <= x: a(n) <= x}. Gabdullin & Iudelevich show that f(x) ~ x/log log x. - Charles R Greathouse IV, Mar 16 2022