A237353 For n=g+h, a(n) is the minimum value of omega(g)+omega(h).

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2

Offset: 2

Author

Lei Zhou, Feb 06 2014

Keywords

• nonn

Comments

omega(g) is defined in A001221.

If Goldbach's conjecture is true, all items with even index of this sequence is less than or equal to 2.

This sequence is defined for n >= 2.

It is conjectured that the maximum value of this sequence is 3.

2=1+1 makes the only zero term of this sequence a(2)=0.

This sequence gets a(n)=1 when n=1+p^k, where p is a prime number and k >= 1.

Examples

			For n=2, 2=1+1. 1 does not have prime factor. So a(2)=0+0=0;
For n=6, 6=1+5.  1 does not have prime factor where 5 has one. Another case 6=3+3 yields sum of prime factors of g and h 1+1=2.  Since 1 < 2, according to the definition, we chose the smaller one. So a(6)=1;
For n=7, 7=2+5.  Both 2 and 5 have one prime factor.  So a(7)=1+1=2;
For n=331, one of the case is 331=2+329=2+7*47.  In which 2 has one prime factor, and 329 has two.  So a(331)=1+2=3.
		

Links

• Lei Zhou, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A002375, A001221

Programs

Mathematica

Table[ct = n; Do[h = n - g; c = Length[FactorInteger[g]] + Length[FactorInteger[h]]; If[g == 1, c--]; If[h == 1, c--]; If[c < ct, ct = c], {g, 1, Floor[n/2]}]; ct, {n, 2, 88}]
Table[ Min@Table[PrimeNu[ n - k ] + PrimeNu[  k  ], {k, n - 1}], {n, 2, 88}]

Sage

def a(n): return min(A001221(a)+A001221(n-a) for a in range(1,floor(n/2)+1)) # Ralf Stephan, Feb 23 2014