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

%I A278293 #27 Nov 30 2016 13:09:43
%S A278293 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,
%T A278293 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,
%U A278293 3,2,2,2,4,3,2,2,3,3,2
%N A278293 a(n) is the number of prime factors of A278291(n) (with multiplicity).
%C A278293 a(n) is also the number of prime factors in A045920(n), by definition.
%C A278293 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.
%H A278293 Ely Golden, <a href="/A278293/b278293.txt">Table of n, a(n) for n = 1..10000</a>
%F A278293 a(n) = A001222(A278291(n)) = A001222(A045920(n))
%e A278293 a(2)=2, as A278291(2)=10, which has 2 prime factors.
%e A278293 a(6)=3, as A278291(6)=28, which has 3 prime factors.
%o A278293 (Java) public class A278293{
%o A278293 public static void main(String[] args)throws Exception{
%o A278293     long dim0=numberOfPrimeFactors(2);//note that this method must be manually implemented by the user
%o A278293     long dim1;
%o A278293     long counter=3;
%o A278293     long index=1;
%o A278293     while(index<=10000){
%o A278293       dim1=numberOfPrimeFactors(counter);
%o A278293       if(dim1==dim0){System.out.println(index+" "+dim1);index++;}
%o A278293       dim0=dim1;
%o A278293       counter++;
%o A278293     }
%o A278293   }
%o A278293 }
%o A278293 (SageMath)
%o A278293 def bigomega(x):
%o A278293     s=0;
%o A278293     f=list(factor(x));
%o A278293     for c in range(len(f)):
%o A278293         s+=f[c][1]
%o A278293     return s;
%o A278293 dim0=bigomega(2);
%o A278293 counter=3
%o A278293 index=1
%o A278293 while(index<=10000):
%o A278293     dim1=bigomega(counter);
%o A278293     if(dim1==dim0):
%o A278293         print(str(index)+" "+str(dim1))
%o A278293         index+=1;
%o A278293     dim0=dim1;
%o A278293     counter+=1;
%Y A278293 Cf. A045920(n).
%K A278293 nonn
%O A278293 1,2
%A A278293 _Ely Golden_, Nov 16 2016