A363927 Numbers N such that in the concatenation of N^2 and N^3, each of the 10 decimal digits appears equally often.

69, 6534, 497375, 539019, 543447, 586476, 589629, 601575, 646479, 858609, 895688, 959097, 46839081, 47469378, 47693199, 47760623, 47841576, 48038964, 48527792, 48733506, 48886836, 48965892, 49229103, 49397283, 49594832, 49670616, 50013116, 50247423, 50359157

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", "))

Crossrefs

Cf. A363905, A363909: concat(n^2, n^3) has each digit at least once / twice.

Cf. A171102: pandigital numbers.

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

Cf. A119735: n^3 is pandigital.

Sequence in context: A207197 A207217 A296785 * A364024 A264171 A297550

Adjacent sequences: A363924 A363925 A363926 * A363928 A363929 A363930

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