OFFSET
1,1
COMMENTS
Choose the first number not leading to a contradiction.
The sequence starting with 1 is hard: {1, 5, 9, 86, 212, ...};
Sequence starting with 3: {3, 7, 19, 32, 55, 246, 39499, ...};
Sequence starting with 4: {4, 6, 29, 89, 137, 749, 1685, 16497, ...}.
EXAMPLE
The subset {2, 4, 31} produces the three sums {6, 33, 35} which factor as {2*3, 3*11, 5*7}.
MAPLE
with(numtheory):nn:=500000:T:=array(1..nn): U:=array(1..nn): for p from 1 to
nn do: T[p]:=p+1:U[p]:=2:od:for u from 1 to 10 do: k:=1+u:for n from u+1 to
nn do:s:=T[n]+T[u]:s1:=nops(factorset(s)):s2:=bigomega(s):if s1=2 and s2=2 then
U[k]:=T[n]:k:=k+1:else fi:od:for i from 1 to nn do:T[i]:=U[i]:od:od:for j from
1 to 30 do:print( T[j]):od:
MATHEMATICA
TwoDistinct[n_]:=Module[{p, e}, {p, e}=Transpose[FactorInteger[n]]; Length[p]==2 && e=={1, 1}]; t={2}; k=2; Do[While[k++; !And@@TwoDistinct/@(k+t)]; AppendTo[t, k], {6}]; t
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Jan 31 2011
EXTENSIONS
Removed 84835 and a(10)-a(12) from Donovan Johnson, Feb 14 2011
STATUS
approved