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 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. 7
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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 LINKS Alois P. Heinz, Table of n, a(n) for n = 1..20000 EXAMPLE 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}... 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 CROSSREFS Cf. A133837. Sequence in context: A055187 A217780 A329316 * A302240 A130307 A130314 Adjacent sequences:  A109408 A109409 A109410 * A109412 A109413 A109414 KEYWORD nonn AUTHOR Zak Seidov, Jul 01 2005 STATUS approved

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Last modified August 15 09:19 EDT 2022. Contains 356135 sequences. (Running on oeis4.)