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

2,6

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.

Links

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

Example

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.

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}]

Prog

(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

Crossrefs

Cf. A002375, A001221

Sequence in context: A302041 A302031 A371883 * A293460 A231813 A158210

Adjacent sequences: A237350 A237351 A237352 * A237354 A237355 A237356

Keyword

nonn

Author

Lei Zhou, Feb 06 2014

Status

approved