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A363927
Numbers N such that in the concatenation of N^2 and N^3, each of the 10 decimal digits appears equally often.
2
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
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
KEYWORD
nonn,base
AUTHOR
EXTENSIONS
a(13) and beyond from Michael S. Branicky, Jun 28 2023
STATUS
approved