A126326 - OEIS

%I #21 Jun 21 2019 12:30:26

%S 1,2,5,9,13,31,35,92,118,280,516,752,1618,1968,5090,6594,15620,19556,

%T 48364,61552,149028,188140,460272,583376,1419928,1796208,4382888,

%U 5549640,13524944,17117360,41741584,52840864,128817168,163051888,397550784,503232512,1226887072

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

%C Analog of A065026, with powers.

%F Conjectures from _Colin Barker_, Jun 21 2019: (Start)

%F G.f.: x*(1 - x)*(1 + 3*x + 6*x^2 + 11*x^3 + 10*x^4 + 15*x^5 + 6*x^6 + 4*x^7 + 10*x^8 + 166*x^10 + 52*x^11 + 236*x^12 - 236*x^13 - 210*x^14) / (1 - 2*x^2 - 4*x^4 + 2*x^6).

%F a(n) = 2*a(n-2) + 4*a(n-4) - 2*a(n-6) for n>16.

%F (End)

%e a(4) = 9 because the possible sums and products of a(1), a(2), a(3) are 1, 2, 5, 1+2, 1+5, 2+5, 1+2+5, 2*5, 2^2, 2^3, ..., 5^2, 5^3, ... = 1, 2, 4, 3, 4, 5, 6, 7, 8, 10, 16, 25, ... The smallest missing number is 9.

%p A126326 := proc(amax) local a,n,sumset,prodset,j,powset,aprev,newsumset,newprodset ; a := [1,2] ; n := 3 ; sumset := {} ; prodset := {1} ; powset := {1} ; while n <= amax do aprev := op(-1,a) ; newsumset := sumset ; for j from 1 to nops(sumset) do if op(j,sumset)+aprev <= amax then newsumset := newsumset union { op(j,sumset)+aprev } ; fi ; od ; for j from 1 to nops(a)-1 do if op(j,a)+aprev <= amax then newsumset := newsumset union { op(j,a)+aprev } ; fi ; od ; sumset := newsumset ; newprodset := prodset ; for j from 1 to nops(prodset) do if op(j,prodset)*aprev <= amax then newprodset := newprodset union { op(j,prodset)*aprev } ; fi ; od ; for j from 1 to nops(a)-1 do if op(j,a)*aprev <= amax then newprodset := newprodset union { op(j,a)*aprev } ; fi ; od ; prodset := newprodset ; for j from 2 to floor(log(amax)/log(aprev)) do if aprev^j <= amax then powset := powset union { aprev^j } ; fi ; od ; while n in sumset or n in prodset or n in powset do n := n+1 ; od ; if n <= amax then a := [op(a),n] ; fi ; print(a) ; n := n+1 ; od ; RETURN(a) ; end: A126326(200000) ; # _R. J. Mathar_, Apr 03 2007

%Y Cf. A065026.

%K nonn

%O 1,2

%A _Jonathan Vos Post_, Mar 11 2007

%E More terms from _R. J. Mathar_, Apr 03 2007

%E a(21)-a(22) from _Nathaniel Johnston_, Oct 02 2011

%E a(23)-a(28) from _Charlie Neder_, Jun 02 2019

%E a(29)-a(37) from _Giovanni Resta_, Jun 03 2019