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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.
3
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
2,3
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.
EXAMPLE
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.
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}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Lei Zhou, Feb 06 2014
STATUS
approved