A377505 a(n) is the number of positive integers that have Omega(n) prime factors and these are all <= n.

1, 1, 2, 3, 3, 6, 4, 20, 10, 10, 5, 35, 6, 21, 21, 126, 7, 84, 8, 120, 36, 36, 9, 495, 45, 45, 165, 165, 10, 220, 11, 3003, 66, 66, 66, 1001, 12, 78, 78, 1365, 13, 455, 14, 560, 560, 105, 15, 11628, 120, 680, 120, 680, 16, 3876, 136, 3876, 136, 136, 17, 4845, 18

Offset: 1

Author

Felix Huber, Dec 20 2024

Keywords

• nonn

Comments

If drawing at random with replacement from the primes <= n as many as n has prime factors, 1/a(n) is the probability that the product of the prime numbers drawn is equal to n.

Examples

			a(4) = 3 because 3 positive integers have Omega(4) = 2 prime factors <= 4: 4 = 2*2, 6 = 2*3, 9 = 3*3.
a(6) = 6 because 6 positive integers have Omega(6) = 2 prime factors <= 6: 4 = 2*2, 6 = 2*3, 9 = 3*3, 10 = 2*5, 15 = 3*5, 25 = 5*5.
a(7) = 4 because 4 positive integers have Omega(7) = 1 prime factor <= 7: 2, 3, 5, 7.
		

Links

• Felix Huber, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Fundamental theorem of arithmetic

Crossrefs

Cf. A000040, A000720, A001222, A007318, A037031, A377537.

Programs

Maple

with(NumberTheory):
A377505:=n->binomial(pi(n)+Omega(n)-1,Omega(n));
seq(A377505(n),n=1..61);

Mathematica

Table[Binomial[PrimePi[n]+PrimeOmega[n]-1,PrimeOmega[n]],{n,61}] (* James C. McMahon, Dec 24 2024 *)

Formula

a(n) = binomial(pi(n) + Omega(n) - 1, Omega(n)) where pi = A000720 and Omega = A001222.

a(p) = pi(p) for prime p.