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

Author

M. F. Hasler, Jun 27 2023

Keywords

• nonn
• base
• less

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.

Examples

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

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10000
• Harold Suarez, Interesting..., Number Theory group on LinkedIn, June 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.

Programs

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 *)

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