A171829 Nonnegative integers that can be made by using six sixes (6 6's) and the four basic operators {+, -, *, /}.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 54, 60, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 84, 90, 96, 102, 107

Offset: 1

Author

Sergio Pimentel, Dec 19 2009

Keywords

• nonn
• fini
• full

Comments

More integers can be made if other operators are allowed (i.e., 22 = 6!/(6*6)+(6+6)/6). The sequence is finite: a(198) = 6*6*6*6*6*6 = 46656 is the last term.

See A258068 ff. for the integers that can be generated with the four basic operators and 7 7's, 8 8's, 9 9's, etc...

Examples

			49 is in the sequence: 49 = (6 + 6/6) * (6 + 6/6).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..198
• Wikipedia, Four Fours

Crossrefs

Cf. A171826, A171827, A171828, A258068, A258069, A258070, A258071.

Cf. A182002, A258097.

Programs

Maple

f:= proc(n) f(n):= `if`(n=1, {6}, {seq(seq(seq([x+y, x-y, x*y,
      `if`(y=0, [][], x/y)][], y=f(n-j)), x=f(j)), j=1..n-1)})
    end:
sort([select(z->z>=0 and is(z, integer), f(6))[]])[];
#  Alois P. Heinz, Aug 04 2013

Mathematica

f[1] = {6}; f[n_] := f[n] = Union @ Flatten @ Table[Table[Table[{x+y, x-y, x*y, If[y == 0, Null, x/y]}, {y, f[n-j]}], {x, f[j]}], {j, 1, n-1}];
Sort[Select[f[6], # >= 0 && IntegerQ[#]&]] (* Jean-François Alcover, Jun 01 2018, after Alois P. Heinz *)

PARI

A171829(n=6, S=Vec([[n]],n))={for(n=2, n, S[n]=Set(concat(vector(n\2, k, concat([concat([[T+U, T-U, U-T, if(U, T/U), if(T, U/T), T*U] | T <- S[k]]) | U <- S[n-k]]))))); select(t-> t>=0 && denominator(t)==1,S[n])} \\ A171829() yields this sequence. Optional args allow to compute variants. - M. F. Hasler, Nov 24 2018

Extensions

Corrected and edited by Alois P. Heinz, Aug 03 2013