A176747 Triangular numbers and numbers which cannot be represented as a sum of two earlier members of the sequence.

0, 1, 3, 5, 6, 10, 14, 15, 21, 23, 28, 32, 36, 40, 45, 52, 55, 66, 74, 78, 82, 86, 91, 105, 113, 117, 120, 124, 136, 153, 155, 166, 171, 184, 190, 197, 201, 209, 210, 217, 228, 231, 247, 253, 276, 278, 300, 311, 325, 349, 351, 378, 390, 406, 435, 439, 465, 474, 496, 516, 518

Offset: 0

Author

Vladimir Shevelev, Apr 25 2010

Keywords

• nonn

Examples

			5 is the smallest number which is not represented as sum of 2 numbers of the set {0,1,3}. Therefore 5 is in the sequence.
14 is the smallest number which is not represented as sum of 2 numbers of the set {0,1,3,5,6,10}. Therefore 14 is in the sequence.
		

Crossrefs

Cf. A000217, A176744, A176745, A176746.

Programs

Maple

isA000217 := proc(n) issqr(8*n+1) ; end proc:
A176747 := proc(n) option remember; if n <=1 then n; else for a from procname(n-1)+1 do if isA000217(a) then return a; end if;
isrep := false; for i from 1 to n-1 do for j from i to n-1 do if procname(i)+procname(j) = a then isrep := true; end if; end do: end do: if not isrep then return a; end if; end do: end if; end proc:
seq(A176747(n),n=0..60) ; # R. J. Mathar, Nov 01 2010
# Alternative:
A176747_list := proc(upto) local P, k, issum, istri; P := [];
issum := k -> ormap(p -> member(k - p, P), P);
istri := k -> issqr(8*k + 1);
for k from 0 to upto do
    if istri(k) or not issum(k) then P := [op(P), k] fi od;
P end: print(A176747_list(518));  # Peter Luschny, Jul 20 2022

Mathematica

A176747list[upto_] := Module[{P = {}, issum, istri},
issum[k_] := AnyTrue[P, MemberQ[P, k-#]&];
istri[k_] := IntegerQ@Sqrt[8k+1];
For[k = 0, k <= upto, k++,
   If[istri[k] || !issum[k], AppendTo[P, k]]];
P];
A176747list[518] (* Jean-François Alcover, Sep 26 2022, after Peter Luschny *)

Extensions

Definition rephrased, sequence extended beyond 55 by R. J. Mathar, Nov 01 2010