A278293 a(n) is the number of prime factors of A278291(n) (with multiplicity).

1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 4, 2, 2, 2, 3, 3, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 2, 3, 3, 4, 2, 2, 2, 2, 3, 2, 4, 3, 2, 3, 4, 2, 3, 2, 2, 3, 3, 3, 3, 3, 2, 2, 2, 4, 3, 2, 2, 3, 3, 2

Offset: 1

Author

Ely Golden, Nov 16 2016

Keywords

• nonn

Comments

a(n) is also the number of prime factors in A045920(n), by definition.

Empirically, it seems as though there are relatively fewer instances of a(n)=x as x tends toward positive infinity (with the exception of a(n)=1, of which there is exactly one instance due to 2 and 3 being the only consecutive primes). For example, in the first 10000 terms, 2391 are 2, 5046 are 3, 2126 are 4, 381 are 5, 51 are 6, 3 are 7, and only one is 8, with no terms in the first 10000 greater than 8.

Examples

			a(2)=2, as A278291(2)=10, which has 2 prime factors.
a(6)=3, as A278291(6)=28, which has 3 prime factors.
		

Links

• Ely Golden, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A045920(n).

Programs

Java

public class A278293{
public static void main(String[] args)throws Exception{
    long dim0=numberOfPrimeFactors(2);//note that this method must be manually implemented by the user
    long dim1;
    long counter=3;
    long index=1;
    while(index<=10000){
      dim1=numberOfPrimeFactors(counter);
      if(dim1==dim0){System.out.println(index+" "+dim1);index++;}
      dim0=dim1;
      counter++;
    }
  }
}

SageMath

def bigomega(x):
    s=0;
    f=list(factor(x));
    for c in range(len(f)):
        s+=f[c][1]
    return s;
dim0=bigomega(2);
counter=3
index=1
while(index<=10000):
    dim1=bigomega(counter);
    if(dim1==dim0):
        print(str(index)+" "+str(dim1))
        index+=1;
    dim0=dim1;
    counter+=1;

Formula

a(n) = A001222(A278291(n)) = A001222(A045920(n))