Interesting Seq ? Partition in groups with semiprime sums

[zak seidov]

Dear seqfan gurus,
i dare submit this sequence with hope
that someone may wish to answer
the Qs in it,
thanks, zak

%I A109411
%S A109411 
3,1,4,1,1,5,2,3,1,1,13,3,1,3,2,2,2,1,4,6,2,1,6,1,2,2,1,14,4,1,1,1,
%N A109411 Partition of sequence of natural numbers in
the minimal 
groups such that sum of terms in each group is a
semiprime.
%C A109411 The partiton starts as this: 
{1-3},{4},{5-8},{9},{10},{11-15},{16-17},{18-20},{21},{22},{23-35},
{36-38},{39},{40-42},{43-44},{45-46},{47-48},{49},{50-53},
{54-59},{60-61},{62},{63-68},{69},{70-71},{72-73},{74},{75-88},
{89-92},{93},{94},{95},{96-98},{99-103},{104-105}...
The sequence A109411 gives the number of terms in each
group. 
The big question is: is the sequence finite? If a
group begins with a 
and ends with b then sum of terms is s=(a+b)(a-b+1)/2
and it's not 
evident that :
a) there are a's such that it's impossible to find
b>=a such that s 
is semiprime,  
b) such a's will appear in A109411. I leave it to
seqfans. I present 
"dirty" Mmca (as is) which works. The longest group
found is {1770-1843} 
consisting of 74 terms.
%t A109411 
s={{1,2,3}};a=4;Do[Do[If[Plus@@Last/@FactorInteger[(a+x)(x-a+1)/2]==2,AppendTo[s,Range[a,x]];(*Print[Range[a,x]];*)a=x+1;Break[]],{x,a,20000}],{k,1,1000}];s
%O A109411 1
%K A109411 ,nice,nonn,unkn,
%A A109411 Zak Seidov (zakseidov at yahoo.com), Jul 01
2005



		
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