

A363905


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


4



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
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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

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


EXAMPLE

69^2 = 4761, 69^3 = 328509, which together contain each digit 09 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)))


CROSSREFS

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


KEYWORD

nonn,base,less


AUTHOR



STATUS

approved



