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A303600 a(1)=2 and a(2)=3, then a(n+1) is the smallest integer larger than a(n) that can be written as the sum of two (not necessarily distinct) earlier terms in exactly one way.

%I A303600 #27 Jun 20 2025 01:24:59
%S A303600 2,3,4,5,9,10,11,16,22,24,28,29,30,37,42,50,55,56,70,73,76,82,89,95,
%T A303600 101,102,115,128,133,135,136,141,142,153,160,161,168,174,181,195,199,
%U A303600 200,214,221,227,233,247,252,265,266,267,273,280,285,286,325,331,332,338
%N A303600 a(1)=2 and a(2)=3, then a(n+1) is the smallest integer larger than a(n) that can be written as the sum of two (not necessarily distinct) earlier terms in exactly one way.
%H A303600 David A. Corneth, <a href="/A303600/b303600.txt">Table of n, a(n) for n = 1..10000</a> (Terms 1..100 from David Consiglio, Jr.)
%H A303600 David A. Corneth, <a href="/A303600/a303600.gp.txt">PARI-prog</a>
%H A303600 Borys Kuca, <a href="https://arxiv.org/abs/1804.09594">Structures in Additive Sequences</a>, arXiv:1804.09594 [math.NT], 2018. See V(2,3).
%t A303600 Nest[Append[#, Function[{m, s}, First@ SelectFirst[Tally[s], And[First@ # > m, Last@ # < 3] &]] @@ {Max@ #, Sort[Total /@ Tuples[#, {2}]]}] &, {2, 3}, 57] (* _Michael De Vlieger_, Apr 27 2018 *)
%o A303600 (PARI) \\ See PARI link \\ _David A. Corneth_, Apr 27 2018
%o A303600 (Python)
%o A303600 terms = [2,3]
%o A303600 while len(terms) < 100:
%o A303600     options = []
%o A303600     for x in range(len(terms)):
%o A303600         for y in range(x,len(terms)):
%o A303600             options.append(terms[x]+terms[y])
%o A303600     for y in sorted(options):
%o A303600         if options.count(y) == 1 and y > max(terms):
%o A303600             terms.append(y)
%o A303600             break
%o A303600 print(terms)
%o A303600 # _David Consiglio, Jr._, Apr 18 2018
%o A303600 (Python)
%o A303600 from collections import Counter
%o A303600 def A303600_list(num_terms):
%o A303600     terms = [2, 3]
%o A303600     sum_counts = Counter([4, 5, 6])
%o A303600     next_term = 3
%o A303600     for n in range(2, num_terms):
%o A303600         next_term += 1
%o A303600         while sum_counts[next_term] != 1: next_term += 1
%o A303600         terms.append(next_term)
%o A303600         for term in terms: sum_counts[term + next_term] += 1
%o A303600     return terms
%o A303600 print(A303600_list(59))
%o A303600 # _David Radcliffe_, Jun 19 2025
%Y A303600 Cf. A004280 (with first terms 1 and 2).
%K A303600 nonn
%O A303600 1,1
%A A303600 _Michel Marcus_, Apr 26 2018
%E A303600 More terms from _David Consiglio, Jr._, Apr 26 2018