OFFSET
1,1
COMMENTS
Any of these integers can be expressed by a composition of floor functions f(n) = [mu*n] and g(n) = [nu*n], provided that the composition applies at least one f(n) and one g(n), for an irrational slope 1 < mu < 2 and its conjugate nu = 1/(1-1/mu). This follows from the Rayleigh-Beatty theorem. See reference in link. A064801 gives "floor squares."
LINKS
EXAMPLE
Example (of calculation by sieve, see reference in link, p. 221): The first term, 2, while given by the (un-nested) floor [mu] of a real slope 2 <= mu < 3, is too big to result from a twice-nested floor [[mu]mu], thrice-nested floor [mu[mu[mu]]], etc. for mu < 2. Yet for mu >= 2, the integer 2 is too small to result from a twice-nested, thrice-nested, etc. floor. Sequence A064801 = 1,4,5,9,... gives the "floor squares" - positive integers that are expressible as the twice-nested floor [mu[mu]] for a positive real slope mu. Thus 2,3,6,7 and 8 are not "floor squares". Besides 0 and 1, the next smallest integer obtainable from nesting a floor function with real positive slope t times is 2^t. Thus, the sequence of positive "floor cubes" starts with 1 and continues 8,9,12,13,14,27,... Hence, the first level of the sieve catches the floor squares 1,4,5,9,..., the second level of the sieve catches the floor cubes 1,8,... So, 2,3,6, and 7 are the initial floor-powerfree numbers passing the sieve for all t >= 2.
MATHEMATICA
(*Define the nested floor function.*)
NestedFloor[slope_, t_] := Nest[Function[Floor[#*slope]], 1, t]
(*Specify an upper bound on a(n) in DATA.*)
aMax = 1017;
(*Calculate the number of floor powers that must be sifted out.*)
tMax = Ceiling[Log[2, aMax]];
(*Initialize slopes for each floor power.*)
slopes = Table[{1}, {tMax}]; slopes[[1]] = Table[n, {n, 1, aMax}];
(*Initialize "floor-powerful" numbers for each floor power.*)
powerfuls = Table[{1}, {tMax}]; powerfuls[[1]] = Table[n, {n, 1, aMax}];
Do[n = 2; While[Last[powerfuls[[t]]] < aMax,
(*Include slopes from previously sifted power as coarse slopes.*) coarseSlope = slopes[[t - 1]][[n]]; AppendTo[slopes[[t]], coarseSlope]; AppendTo[powerfuls[[t]], NestedFloor[coarseSlope, t]];
(*Generate fine slopes between the coarse slopes; use floor-powerful numbers from previously sifted floor power as denominators q, start with a numerator p that gives the least fine slope exceeding the current coarse one*) q = powerfuls[[t - 1]][[n]]; p = Floor[coarseSlope*q] + 1; fineSlope = p/q;
(*Insert fine slope(s) (if any) between the current coarse slope and the next smallest one.*) nextCoarse = slopes[[t - 1]][[n + 1]]; While[fineSlope < nextCoarse, AppendTo[slopes[[t]], fineSlope]; AppendTo[powerfuls[[t]], NestedFloor[fineSlope, t]]; p++; fineSlope = p/q; ]; n++], {t, 2, tMax}]
(*Sift out all floor-powerful numbers to output the floor-powerfree numbers, a(n)*)
Complement[Table[n, {n, 1, aMax}], Union[Flatten[Rest[powerfuls]]]]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
J. Parker Shectman, Apr 06 2021
STATUS
approved