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A005243 A self-generating sequence: start with 1 and 2, take all sums of any number of successive previous elements and adjoin them to the sequence. Repeat!

%I A005243 M0623 #49 Aug 04 2025 04:10:55
%S A005243 1,2,3,5,6,8,10,11,14,16,17,18,19,21,22,24,25,29,30,32,33,34,35,37,40,
%T A005243 41,43,45,46,47,49,51,54,57,58,59,60,62,65,67,68,69,70,71,72,73,75,76,
%U A005243 77,78,80,81,82,84,86,87,88,90,91,92,93,94,95,96,97,99,100
%N A005243 A self-generating sequence: start with 1 and 2, take all sums of any number of successive previous elements and adjoin them to the sequence. Repeat!
%C A005243 Most of the natural numbers are members. Conjecture: there are infinitely many nonmembers. Is there an estimate for a(k)/k ?
%C A005243 A118164(n) = number of representations of a(n) as sum of consecutive earlier terms. - _Reinhard Zumkeller_, Apr 13 2006
%D A005243 R. K. Guy, Unsolved Problems in Number Theory, E31.
%D A005243 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005243 T. D. Noe, <a href="/A005243/b005243.txt">Table of n, a(n) for n=1..1000</a>
%H A005243 Thomas Bloom, <a href="https://www.erdosproblems.com/423">Problem 423</a>, Erdős Problems.
%H A005243 D. R. Hofstadter, <a href="/A006336/a006336_1.pdf">Eta-Lore</a> [Cached copy, with permission]
%H A005243 D. R. Hofstadter, <a href="/A006336/a006336_2.pdf">Pi-Mu Sequences</a> [Cached copy, with permission]
%H A005243 D. R. Hofstadter and N. J. A. Sloane, <a href="/A006336/a006336.pdf">Correspondence, 1977 and 1991</a>
%H A005243 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HofstadterSequences.html">Hofstadter Sequences.</a>
%e A005243 After 1,2,3,5,6 you can adjoin 8 = 3+5, 10 = 2+3+5, etc.
%e A005243 12 is not a term since it is not the sum of any set of consecutive previous terms.
%t A005243 nmax = 200; For[ s = {1, 2}; n = 3, n <= nmax, n++, ls = Length[s]; tt = Total /@ Flatten[Table[s[[i ;; j]], {i, 1, ls-1}, {j, i+1, ls}], 1]; If[MemberQ[tt, n], AppendTo[s, n]]]; A005243 = s (* _Jean-François Alcover_, Oct 21 2016 *)
%o A005243 (Haskell)
%o A005243 import Data.Set (singleton, deleteFindMin, fromList, union, IntSet)
%o A005243 a005243 n = a005243_list !! (n-1)
%o A005243 a005243_list = 1 : h [1] (singleton 2) where
%o A005243    h xs s = m : h (m:xs) (union s' $ fromList $ map (+ m) $ scanl1 (+) xs)
%o A005243      where (m, s') = deleteFindMin s
%o A005243 -- _Reinhard Zumkeller_, Dec 17 2015, Jun 22 2011
%Y A005243 Complement of A048973.
%Y A005243 Cf. A118065, A118166.
%K A005243 nonn,nice,easy
%O A005243 1,2
%A A005243 D. R. Hofstadter, Jul 15 1977
%E A005243 More terms from _Jud McCranie_