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

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, 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, 5, 4, 6, 4, 5, 5, 5, 5, 5, 4, 5, 5, 5, 4, 6, 5, 5, 5, 5

Offset: 2

Author

Lei Zhou, Feb 06 2014

Keywords

• nonn
• easy

Comments

omega(g) is defined in A001221.

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).

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.

Examples

			For n=2, 2=1+1. 1 does not have prime factor. So a(2)=0+0=0;
For n=3, 3=1+2, 1 does not have prime factor, where 2 has one. So a(3)=0+1=1;
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.
...
For n=211, in best case we have 211=105+106=3*5*7+2*53.  So a(211)=3+2=5.
		

Links

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

Crossrefs

Cf. A237353, A002375, A001221.

Programs

Maple

A237354 := proc(n)
    local a,g,om ;
    a := 0 ;
    for g from 1 to n/2 do
        om := A001221(g)+A001221(n-g) ;
        if om > a then
            a := om ;
        end if;
    end do:
    a ;
end proc:
seq(A237354(n),n=2..100) ; # R. J. Mathar, Feb 13 2014

Mathematica

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}]
(* Wouter Meeussen : *) Table[ Max@Table[PrimeNu[ n - k ] + PrimeNu[  k  ], {k, n - 1}], {n, 2, 88}]