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! (Formerly M0623)

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, 41, 43, 45, 46, 47, 49, 51, 54, 57, 58, 59, 60, 62, 65, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 80, 81, 82, 84, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 97, 99, 100

Offset

1,2

Comments

Most of the natural numbers are members. Conjecture: there are infinitely many nonmembers. Is there an estimate for a(k)/k ?

A118164(n) = number of representations of a(n) as sum of consecutive earlier terms. - Reinhard Zumkeller, Apr 13 2006

References

R. K. Guy, Unsolved Problems in Number Theory, E31.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n=1..1000

Thomas Bloom, Problem 423, Erdős Problems.

D. R. Hofstadter, Eta-Lore [Cached copy, with permission]

D. R. Hofstadter, Pi-Mu Sequences [Cached copy, with permission]

D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991

Eric Weisstein's World of Mathematics, Hofstadter Sequences.

Example

After 1,2,3,5,6 you can adjoin 8 = 3+5, 10 = 2+3+5, etc.
12 is not a term since it is not the sum of any set of consecutive previous terms.

Mathematica

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 *)

Prog

(Haskell)
import Data.Set (singleton, deleteFindMin, fromList, union, IntSet)
a005243 n = a005243_list !! (n-1)
a005243_list = 1 : h [1] (singleton 2) where
   h xs s = m : h (m:xs) (union s' $ fromList $ map (+ m) $ scanl1 (+) xs)
     where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Dec 17 2015, Jun 22 06 2011

Crossrefs

Complement of A048973.

Cf. A118065, A118166.

Sequence in context: A248560 A179180 A085921 * A117045 A324793 A244053

Adjacent sequences: A005240 A005241 A005242 * A005244 A005245 A005246

Keyword

nonn,nice,easy

Author

D. R. Hofstadter, Jul 15 1977

Extensions

More terms from Jud McCranie

Status

approved