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A237354 a(n) is the maximum of omega(g)+omega(h) for all decompositions n=g+h with g>=h>=1.

%I A237354 #18 Feb 14 2014 10:13:28
%S A237354 0,1,2,2,2,2,3,3,3,3,4,3,3,3,4,3,4,3,4,4,4,3,4,4,4,4,4,4,4,4,4,4,4,4,
%T A237354 5,4,4,4,5,4,5,4,5,5,4,4,5,4,5,5,5,4,5,4,5,5,5,4,6,4,5,5,5,5,5,4,5,5,
%U A237354 5,4,6,4,5,5,5,5,5,4,5,5,5,4,6,5,5,5,5
%N A237354 a(n) is the maximum of omega(g)+omega(h) for all decompositions n=g+h with g>=h>=1.
%C A237354 omega(g) is defined in A001221.
%C A237354 The smallest n that makes a(n)=2k should be twice the product of the first k-th prime numbers.  For example, a(4)=2, 4=2*2; a(12)=4, 12=2*(2*3); a(60)=6, 60=2*(2*3*5).
%C A237354 The largest n that makes a(n)=k should be smaller than or equal to the product of the first k-th primes plus 1. For example, a(3)=1, 3 = 2+1; a(7)=2, 7=2*3+1; a(23)=3, 23<2*3*5+1=31; a(89)=4, 89<211=2*3*5*7+1.
%H A237354 Lei Zhou, <a href="/A237354/b237354.txt">Table of n, a(n) for n = 2..10000</a>
%e A237354 For n=2, 2=1+1. 1 does not have prime factor. So a(2)=0+0=0;
%e A237354 For n=3, 3=1+2, 1 does not have prime factor, where 2 has one. So a(3)=0+1=1;
%e A237354 For n=4, 4=1+3=2+2.  From 1+3 we got 1, from 2+2 we got 2.  The larger one is 2.  So a(4)=1+1=2.
%e A237354 ...
%e A237354 For n=211, in best case we have 211=105+106=3*5*7+2*53.  So a(211)=3+2=5.
%p A237354 A237354 := proc(n)
%p A237354     local a,g,om ;
%p A237354     a := 0 ;
%p A237354     for g from 1 to n/2 do
%p A237354         om := A001221(g)+A001221(n-g) ;
%p A237354         if om > a then
%p A237354             a := om ;
%p A237354         end if;
%p A237354     end do:
%p A237354     a ;
%p A237354 end proc:
%p A237354 seq(A237354(n),n=2..100) ; # _R. J. Mathar_, Feb 13 2014
%t A237354 Table[ct = 0; 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 A237354 (* _Wouter Meeussen_ : *) Table[ Max@Table[PrimeNu[ n - k ] + PrimeNu[  k  ], {k, n - 1}], {n, 2, 88}]
%Y A237354 Cf. A237353, A002375, A001221.
%K A237354 nonn,easy
%O A237354 2,3
%A A237354 _Lei Zhou_, Feb 06 2014