A325134 a(1) = 1; a(n) = number of prime factors of n counted with multiplicity plus the largest prime index of n.

1, 2, 3, 3, 4, 4, 5, 4, 4, 5, 6, 5, 7, 6, 5, 5, 8, 5, 9, 6, 6, 7, 10, 6, 5, 8, 5, 7, 11, 6, 12, 6, 7, 9, 6, 6, 13, 10, 8, 7, 14, 7, 15, 8, 6, 11, 16, 7, 6, 6, 9, 9, 17, 6, 7, 8, 10, 12, 18, 7, 19, 13, 7, 7, 8, 8, 20, 10, 11, 7, 21, 7, 22, 14, 6, 11, 7, 9, 23

Offset: 1

Author

Gus Wiseman, Apr 02 2019

Keywords

• nonn

Comments

A prime index of n is a number m such that prime(m) divides n.

Also one plus the size of the largest hook contained in the Young diagram of the integer partition with Heinz number n. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Crossrefs

Cf. A000720, A001222, A052126, A056239, A061395, A093641, A112798, A252464, A257990, A325133, A325135.

Programs

Maple

with(numtheory):
a:= n-> `if`(n=1, 1, bigomega(n)+pi(max(factorset(n)[]))):
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 03 2019

Mathematica

Table[If[n==1,1,PrimeOmega[n]+PrimePi[FactorInteger[n][[-1,1]]]],{n,100}]

Formula

a(n) = A001222(n) + A061395(n).

a(n) = A252464(n) + 1.