A006844 a(1)=4, a(2)=5; thereafter a(n) is smallest number that is greater than a(n-1) and having a unique representation as a(j) + a(k) for j

4, 5, 9, 13, 14, 17, 19, 21, 24, 25, 27, 35, 37, 43, 45, 47, 57, 67, 69, 73, 77, 83, 93, 101, 105, 109, 113, 115, 123, 125, 133, 149, 153, 163, 173, 197, 201, 205, 209, 211, 213, 217, 219, 227, 229, 235, 237, 239

Offset: 1

Author

N. J. A. Sloane

Keywords

• easy
• nonn
• nice

Comments

This is the 1-additive sequence with base {4,5}. Apart from three extra terms (4, 14, 24) in the initial segment, this breaks up naturally into segments of 32 terms each. [Finch, 1992]. - N. J. A. Sloane, Aug 12 2015

An Ulam-type sequence - see A002858 for many further references, comments, etc. - T. D. Noe, Jan 21 2008

References

• Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 145-151.
• R. K. Guy, "s-Additive sequences," preprint, 1994.
• R. K. Guy, Unsolved Problems in Number Theory, Section C4.
• 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
• Experimental Mathematics, Home Page
• Steven R. Finch, Stolarsky-Harborth Constant [Broken link]
• Steven R. Finch, Stolarsky-Harborth Constant [From the Wayback machine]
• Steven R. Finch, Patterns in 1-additive sequences, Experimental Mathematics 1 (1992), 57-63.
• R. K. Guy, s-Additive sequences, Preprint, 1994. (Annotated scanned copy)
• R. Queneau, Sur les suites s-additives, J. Combin. Theory, A12 (1972), 31-71.
• Eric Weisstein's World of Mathematics, Ulam Sequence
• Wikipedia, Ulam number
• Index entries for Ulam numbers

Programs

Haskell

a006844 n = a006844_list !! (n-1)
a006844_list = 4 : 5 : ulam 2 5 a006844_list
-- Function ulam as defined in A002858.
-- Reinhard Zumkeller, Nov 03 2011

Mathematica

s = {4, 5}; n0 = 9; dn = 32; m = 192; Do[ AppendTo[s, n = Last[s]; While[n++; Length[ DeleteCases[ Intersection[s, n - s], n/2, 1, 1]] != 2]; n], {n0 + dn}]; Clear[a]; a[n_] := a[n] = If[n <= n0 + dn, s[[n]], a[n - dn] + m]; Table[a[n], {n, 1, 200}] (* Jean-François Alcover, Apr 03 2013 *)

Formula

For n>9, a(n+32) = a(n) + 192. - T. D. Noe, Jan 21 2008