A333300 Numbers that are of the form abab in some base (a <> b, a <> 0).

10, 30, 50, 60, 68, 70, 102, 119, 130, 136, 153, 182, 187, 204, 208, 221, 222, 234, 238, 260, 286, 296, 333, 338, 350, 364, 370, 390, 407, 416, 442, 444, 450, 481, 494, 500, 520, 546, 550, 555, 572, 592, 598, 600, 629, 650, 666, 700, 703, 715, 738, 740, 750

Offset

1,1

Comments

The first terms with 2, 3, ..., 6 representations are 520, 4930, 117130, 111270100, and 3142012250. - Giovanni Resta, Mar 15 2020

Links

Richárd Pintér, Table of n, a(n) for n = 1..10000

Richárd Pintér, Haskell source

Example

10 = 1010_2 is a term, 30 = 1010_3 and 50 = 1212_3 also.

Mathematica

bxy[n_] := Block[{x, y, b, s, bb = Select[Sqrt[ Divisors[n] - 1], IntegerQ[#] && # > 1 && (1 + #^2) # <= n && #^4-#^2-2 >= n &]}, Flatten[ Table[s = Solve[(1 + b^2) (b x + y) == n && 0<x<b && 0<=y<b && x!=y, {x, y}, Integers]; If[s == {}, {}, {b, x, y} /. s], {b, bb}], 1]]; Select[ Range@ 750, bxy[#] != {} &] (* Giovanni Resta, Mar 14 2020 *)

Prog

(Python)
def ok(n):
    base = 2
    while True:
        base3, base2 = base**3, base**2
        if base3 + base > n: return False
        for a in range(1, base):
            for b in range(base):
                if a == b: continue
                t = a*(base3 + base) + b*(base2 + 1)
                if t == n: return True
                elif t > n: break
        base += 1
print([k for k in range(751) if ok(k)]) # Michael S. Branicky, Oct 30 2021

Crossrefs

Sequence in context: A362047 A326122 A027183 * A096844 A031299 A124164

Adjacent sequences: A333297 A333298 A333299 * A333301 A333302 A333303

Keyword

nonn,base

Author

Richárd Pintér, Mar 14 2020

Status

approved