A389643 Squares that are the concatenation of a square and a nonzero cube.

361, 9216, 49729, 168921, 519841, 5184729, 9884736, 259081216, 361000000, 749609641, 1404225729, 8122515625, 9216000000, 49729000000, 70174129216, 89870446656, 147476736729, 168921000000, 519841000000, 745293796416, 1080936581761, 1971364826809, 5184729000000, 9884736000000, 39945218769729

Offset

1,1

Comments

The sequence includes y^2 when (x,y) is a positive integer solution of the Pell equation 10*x^2 + 1 = y^2 (see A023111).

If k is a term, then so is 10^6 * k.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..73

Example

a(3) = 49729 is a term because 49729 = 223^2 is a square and is the concatenation of 49 = 7^2 and 729 = 9^3.

Maple

filter:= proc(n) local i, y;
   for i from 1 to ilog10(n) do
      y:= n mod 10^i;
      if y >= 10^(i-1) and  issqr((n-y)/10^i) and iscube(y) then return true fi
   od;
   false
end proc;
select(filter, [seq(i^2, i=1..10^6)]);

Prog

(Python)
from itertools import count, islice
from sympy import integer_nthroot
def A389643_gen(): # generator of terms
    for x in count(1):
        x2 = x**2
        s = str(x2)
        for i in range(1, len(s)):
            y, z = int(s[:i]), int(s[i:])
            if s[i]>'0' and integer_nthroot(y, 2)[1] and integer_nthroot(z, 3)[1]:
                yield x2
                break
A389643_list = list(islice(A389643_gen(), 20)) # Chai Wah Wu, Oct 14 2025

Crossrefs

Cf. A023111, A039686.

Sequence in context: A303257 A253675 A206249 * A231996 A302385 A303106

Adjacent sequences: A389640 A389641 A389642 * A389644 A389645 A389646

Keyword

nonn,base

Author

Robert Israel, Oct 09 2025

Status

approved