login
A333300
Numbers that are of the form abab in some base (a <> b, a <> 0).
1
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, 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
KEYWORD
nonn,base
AUTHOR
Richárd Pintér, Mar 14 2020
STATUS
approved