A363905 Numbers whose square and cube taken together contain each decimal digit.

69, 128, 203, 302, 327, 366, 398, 467, 542, 591, 593, 598, 633, 643, 669, 690, 747, 759, 903, 923, 943, 1016, 1018, 1027, 1028, 1043, 1086, 1112, 1182, 1194, 1199, 1233, 1278, 1280, 1282, 1328, 1336, 1364, 1396, 1419, 1459, 1463, 1467, 1472, 1475

Offset

1,1

Comments

The first term, a(1) = 69, is the only number for which the square and the cube together contain each decimal digit 0 to 9 exactly once.

a(820) = 6534 is the only number of which the square and cube taken together contain each digit 0 to 9 exactly twice.

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Harold Suarez, Interesting..., Number Theory group on LinkedIn, June 2023.

Example

69^2 = 4761, 69^3 = 328509, which together contain each digit 0-9 exactly once.

Mathematica

fQ[n_] := Union[ Join[ IntegerDigits[n^2], IntegerDigits[n^3]]] == {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}; Select[Range@1500, fQ] (* Robert G. Wilson v, Jun 27 2023 *)

Prog

(PARI) is(k)=#setunion(Set(digits(k^2)), Set(digits(k^3)))>9
select(is, [1..9999])
(Python)
from itertools import count, islice
def A363905_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:len(set(str(n**2))|set(str(n**3)))==10, count(max(startvalue, 1)))
A363905_list = list(islice(A363905_gen(), 20)) # Chai Wah Wu, Jun 27 2023

Crossrefs

Cf. A036744, A054038, A071519 and A156977 for "pandigital" squares.

Cf. A119735: Numbers n such that every digit occurs at least once in n^3.

Sequence in context: A004237 A004238 A039541 * A044192 A044573 A063355

Adjacent sequences: A363902 A363903 A363904 * A363906 A363907 A363908

Keyword

nonn,base,less

Author

M. F. Hasler, Jun 27 2023

Status

approved