OFFSET
1,1
COMMENTS
a(3) = 497375 and a(11) = 895688 are the only terms < 10^6 that are not divisible by 3.
Each term has an even number of decimal digits, k, and a corresponding value between 10^(k-1)*100^(1/3) and 10^k. - Michael S. Branicky, Jun 29 2023
Indeed, the number of digits of concat(N^2, N^3) is floor(2*L + 1) + floor(3*L + 1) where L = log_10(N). This is a multiple of 10 iff L mod 2 is in the interval [5/3, 2), which means that N is in the above range for some even k. - M. F. Hasler, Jul 02 2023
LINKS
Michael S. Branicky, Table of n, a(n) for n = 1..10000
MATHEMATICA
fQ[n_] := Length@ Union[ Count[ Sort[ Join[ IntegerDigits[n^2], IntegerDigits[n^3]]], #] & /@ Range[0, 9]] == 1; Select[ Range@ 52000000, fQ] (* Robert G. Wilson v, Jul 01 2023 *)
PROG
(PARI) is(n)={my(v=concat(digits(n^2), digits(n^3)), c=#v); c%10==0 && vecsort(v)==[0..c-1]\(c\10)}
for(n=1, 1e6, is(n)&& print1(n", "))
KEYWORD
nonn,base
AUTHOR
Charles R Greathouse IV and M. F. Hasler, Jun 28 2023
EXTENSIONS
a(13) and beyond from Michael S. Branicky, Jun 28 2023
STATUS
approved