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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.

%I A363927 #35 Feb 26 2024 09:17:35
%S A363927 69,6534,497375,539019,543447,586476,589629,601575,646479,858609,
%T A363927 895688,959097,46839081,47469378,47693199,47760623,47841576,48038964,
%U A363927 48527792,48733506,48886836,48965892,49229103,49397283,49594832,49670616,50013116,50247423,50359157
%N A363927 Numbers N such that in the concatenation of N^2 and N^3, each of the 10 decimal digits appears equally often.
%C A363927 a(3) = 497375 and a(11) = 895688 are the only terms < 10^6 that are not divisible by 3.
%C A363927 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
%C A363927 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
%H A363927 Michael S. Branicky, <a href="/A363927/b363927.txt">Table of n, a(n) for n = 1..10000</a>
%t A363927 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 *)
%o A363927 (PARI) is(n)={my(v=concat(digits(n^2),digits(n^3)), c=#v); c%10==0 && vecsort(v)==[0..c-1]\(c\10)}
%o A363927 for(n=1,1e6, is(n)&& print1(n","))
%Y A363927 Cf. A363905, A363909: concat(n^2, n^3) has each digit at least once / twice.
%Y A363927 Cf. A171102: pandigital numbers.
%Y A363927 Cf. A036744, A054038, A071519 and A156977 for "pandigital squares".
%Y A363927 Cf. A119735: n^3 is pandigital.
%K A363927 nonn,base
%O A363927 1,1
%A A363927 _Charles R Greathouse IV_ and _M. F. Hasler_, Jun 28 2023
%E A363927 a(13) and beyond from _Michael S. Branicky_, Jun 28 2023