cp's OEIS Frontend

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

%I A237353 #14 Feb 28 2017 22:59:30
%S A237353 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,
%T A237353 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,
%U A237353 2,2,1,2,1,2,2,2,2,2,1,2,1,2,1,2,2,2,2
%N A237353 For n=g+h, a(n) is the minimum value of omega(g)+omega(h).
%C A237353 omega(g) is defined in A001221.
%C A237353 If Goldbach's conjecture is true, all items with even index of this sequence is less than or equal to 2.
%C A237353 This sequence is defined for n >= 2.
%C A237353 It is conjectured that the maximum value of this sequence is 3.
%C A237353 2=1+1 makes the only zero term of this sequence a(2)=0.
%C A237353 This sequence gets a(n)=1 when n=1+p^k, where p is a prime number and k >= 1.
%H A237353 Lei Zhou, <a href="/A237353/b237353.txt">Table of n, a(n) for n = 2..10000</a>
%e A237353 For n=2, 2=1+1. 1 does not have prime factor. So a(2)=0+0=0;
%e A237353 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;
%e A237353 For n=7, 7=2+5.  Both 2 and 5 have one prime factor.  So a(7)=1+1=2;
%e A237353 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.
%t A237353 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}]
%t A237353 Table[ Min@Table[PrimeNu[ n - k ] + PrimeNu[  k  ], {k, n - 1}], {n, 2, 88}]
%o A237353 (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
%Y A237353 Cf. A002375, A001221
%K A237353 nonn
%O A237353 2,6
%A A237353 _Lei Zhou_, Feb 06 2014