A065026 a(1) = 1; for n>1, a(n) = smallest number that is not a sum or product of any subset of the numbers a(1) to a(n-1).

1, 2, 4, 9, 17, 25, 59, 67, 176, 226, 578, 628, 1340, 1876, 4472, 5728, 13952, 17704, 43112, 54568, 133088, 168496, 410720, 519856, 1267568, 1604560, 3911840, 4951552, 12072512, 15281632, 37257248, 47160352, 114980864, 145544128, 354845696, 449166400, 1095100352, 1386188608

Offset: 1

Author

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Nov 02 2001

Keywords

• nonn
• nice

Examples

			a(4) = 9 because the possible sums and products of one or more terms from a(1), a(2), a(3) are 1, 2, 4, 1+2, 1+4, 2+4, 1+2+4, 1, 2, 4, 1*2, 1*4, 2*4, 1*2*4 = 1, 2, 4, 3, 5, 6, 7, 1, 2, 4, 2, 4, 8, 8. The smallest missing number is 9.
		

Programs

Mathematica

(* This program is not efficient for a large number of terms *) a[1] = 1; a[n_] := a[n] = For[s = Subsets[Array[a, n-1]]; sums = Plus @@@ s; prods = Times @@@ s; k = a[n-1]+1, True, k++, If[FreeQ[sums, k], If[FreeQ[prods, k],  Print["a(", n, ") = ", k ]; Return[k]]]]; Array[a, 20] (* Jean-François Alcover, May 26 2015 *)

Formula

Conjectures from Colin Barker, Jun 19 2019: (Start)

G.f.: x*(1 - x)*(1 + 3*x + 5*x^2 + 10*x^3 + 15*x^4 + 14*x^5 + 25*x^6 + 10*x^7 + 8*x^8 + 18*x^9 + 42*x^10 - 402*x^12 - 552*x^13 - 720*x^14 - 804*x^15) / (1 - 2*x^2 - 4*x^4 + 2*x^6).

a(n) = 2*a(n-2) + 4*a(n-4) - 2*a(n-6) for n>17.

(End)

Extensions

More terms from Jon E. Schoenfield, Jun 17 2007

a(35)-a(38) from Donovan Johnson, Jan 11 2011