A109411 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.

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, 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, 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

Offset: 1

Author

Zak Seidov, Jul 01 2005

Keywords

• nonn

Comments

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.

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

Examples

			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}...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A133837.

Programs

Maple

s:= proc(n) option remember; `if`(n<1, 0, a(n)+s(n-1)) end:
a:= proc(n) option remember; local i,k,t; k:=0; t:=s(n-1);
      for i from 1+t do k:=k+i;
        if numtheory[bigomega](k)=2 then return i-t fi
      od
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 26 2015

Mathematica

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