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A122537 a(1) = 1; for n>1, a(n) is smallest number greater than a(n-1), divisible by n and not equal to any a(i)+a(j) with i and j <= n-1.

%I A122537 #11 Oct 11 2019 13:59:34
%S A122537 1,4,6,16,25,30,35,40,45,100,110,120,143,154,180,192,204,216,228,260,
%T A122537 294,330,345,480,500,572,594,616,638,720,744,768,858,884,945,1008,
%U A122537 1036,1102,1131,1160,1189,1218,1247,1320,1395,1426,1457,1584,1617,1700,1734
%N A122537 a(1) = 1; for n>1, a(n) is smallest number greater than a(n-1), divisible by n and not equal to any a(i)+a(j) with i and j <= n-1.
%C A122537 The definition: "a(1) = 1; for n>1, a(n) is smallest number greater than a(n-1) and not equal to any a(i)+a(j) with i and j <= n-1" produces the odd numbers 1, 3, 5, ...
%C A122537 _Jonathan Vos Post_ asks if 1, 2, 4 and 5 are the only values of n for which n^2 divides a(n), Sep 19 2006. _J. Lowell_, Oct 02 2006 remarks that n = 1, 2, 4, 5 and 10 have this property and conjectures that there are no other values.
%e A122537 The 5th term cannot be 20 because 20 = 16+4 and 16 and 4 are both in the sequence.
%p A122537 # a[n] = n-th term of sequence, m[n] = a[n]/n = A122543(n) (Maple program from _N. J. A. Sloane_)
%p A122537 a:=array(0..100000); m:=array(0..100000); hit:=array(0..100000); B:=100000; M:=100;
%p A122537 for n from 1 to B do hit[n]:=0; od:
%p A122537 a[1]:=1; m[1]:=1; a[2]:=4; m[2]:=2; hit[2]:=1; hit[5]:=1; hit[8]:=1;
%p A122537 for n from 3 to M do i:=n*(floor(a[n-1]/n))+n;
%p A122537 while hit[i] = 1 do i:=i+n; od;
%p A122537 a[n]:= i; m[n]:= i/n;
%p A122537 for j from 1 to n do hit[a[j]+i]:=1; od: od:
%p A122537 [seq(a[n],n=1..M)]; [seq(m[n],n=1..M)];
%t A122537 f[s_] := Block[{n, k},n = Length[s] + 1;k = Last[s] + n - Mod[Last[s], n];While[MemberQ[Union[Plus @@@ Tuples[s, 2]], k], k += n];Append[s, k]];Nest[f, {1}, 51] (* _Ray Chandler_, Sep 29 2006 *)
%Y A122537 Cf. A122543 (a(n)/n), A002858, A122544, A122545, A122804.
%K A122537 nonn
%O A122537 1,2
%A A122537 _J. Lowell_, Sep 18 2006
%E A122537 More terms from _N. J. A. Sloane_ and Chai Tian (Chao.Tian(AT)epfl.ch), Sep 19 2006