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%I #13 Nov 26 2015 18:54:32
%S 3,1,4,1,1,5,2,3,1,1,13,3,1,3,2,2,2,1,4,6,2,1,6,1,2,2,1,14,4,1,1,1,3,
%T 5,2,1,2,2,1,3,1,10,2,7,5,4,2,1,2,2,2,6,1,2,3,5,2,3,4,5,6,2,3,2,2,4,1,
%U 14,1,1,4,7,5,2,3,6,1,2,2,2,1,2,2,1,4,2,2,2,3,17,2,3,1,10,3,1,3,6,1,4,2,1
%N Partition the sequence of positive integers into minimal groups so that sum of terms in each group is a semiprime; sequence gives sizes of the groups.
%C Is the sequence finite? If a group begins with a and ends with b then sum of terms is s=(a+b)(b-a+1)/2 and it is not evident that a) there are a's such that it is impossible to find b>=a such that s is semiprime, b) such a's will appear in A109411.
%C The question is equivalent to the following: Given an odd integer n (=2a-1), can it be represented as p-2q or 2q-p where p,q are prime? I believe the answer is "yes" but the problem may have the same complexity as the Goldbach conjecture. - _Max Alekseyev_, Jul 01 2005
%H Alois P. Heinz, <a href="/A109411/b109411.txt">Table of n, a(n) for n = 1..20000</a>
%e The partition begins {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}...
%p s:= proc(n) option remember; `if`(n<1, 0, a(n)+s(n-1)) end:
%p a:= proc(n) option remember; local i,k,t; k:=0; t:=s(n-1);
%p for i from 1+t do k:=k+i;
%p if numtheory[bigomega](k)=2 then return i-t fi
%p od
%p end:
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Nov 26 2015
%t 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
%Y Cf. A133837.
%K nonn
%O 1,1
%A _Zak Seidov_, Jul 01 2005
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