A080256 Sum of numbers of distinct and of all prime factors of n.

0, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 5, 2, 4, 4, 5, 2, 5, 2, 5, 4, 4, 2, 6, 3, 4, 4, 5, 2, 6, 2, 6, 4, 4, 4, 6, 2, 4, 4, 6, 2, 6, 2, 5, 5, 4, 2, 7, 3, 5, 4, 5, 2, 6, 4, 6, 4, 4, 2, 7, 2, 4, 5, 7, 4, 6, 2, 5, 4, 6, 2, 7, 2, 4, 5, 5, 4, 6, 2, 7, 5, 4, 2, 7, 4, 4, 4, 6, 2, 7, 4, 5, 4, 4, 4, 8, 2, 5, 5, 6, 2, 6, 2, 6, 6

Offset

1,2

Comments

a(n) = 2 iff n is prime, A000040; a(n) > 2 iff n is composite, A002808; a(n) <= 3 iff n is prime or square of prime, A000430; a(n) = 3 iff n is square of prime, A001248; a(A080257(n)) > 3;

a(n) <= 4 iff product of proper divisors <= n^2, A007964; a(n) = 4 iff n has four divisors, A030513; a(n) > 4 iff product of proper divisors > n^2, A058080; a(A064598(n)) <= 5; a(A080258(n)) = 5.

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Formula

a(n) = Omega(n) + omega(n) = A001221(n) + A001222(n).

Additive with a(p^e) = e + 1.

Sum_{k=1..n} a(k) = 2 * n * log(log(n)) + c * n + O(n/log(n)), where c = A077761 + A083342 = 1.29615109474508069537... . - Amiram Eldar, Sep 28 2023

Mathematica

f[n_] := Plus @@ (Last /@ FactorInteger[n] + 1); Table[ f[n], {n, 105}] (* Robert G. Wilson v, Aug 03 2005 *)

Prog

(PARI) a(n) = {my(f = factor(n)); omega(f) + bigomega(f); } \\ Amiram Eldar, Sep 28 2023

Crossrefs

Cf. A000040, A000430, A002808, A001248, A007964, A058080, A064598, A080257, A080258.

Cf. A001221, A001222, A046660, A077761, A083342.

Sequence in context: A084302 A289872 A301855 * A351394 A327527 A337454

Adjacent sequences: A080253 A080254 A080255 * A080257 A080258 A080259

Keyword

nonn,easy

Author

Reinhard Zumkeller, Feb 10 2003

Status

approved