%I #76 Feb 09 2024 10:09:25
%S 0,1,4,5,6,9,24,25,49,51,75,76,99,125,249,251,375,376,499,501,624,625,
%T 749,751,875,999,1249,3751,4375,4999,5001,5625,6249,8751,9375,9376,
%U 9999,18751,31249,40625,49999,50001,59375,68751,81249,90624,90625
%N Trimorphic numbers: n^3 ends with n. Also m-morphic numbers for all m > 5 such that m-1 is not divisible by 10 and m == 3 (mod 4).
%C n is in this sequence iff it occurs in one of A002283, A007185, A016090, A198971, A199685, A216092, A216093, A224473, A224474, A224475, A224476, A224477, and A224478. - _Eric M. Schmidt_, Apr 08 2013
%C Let q(n) = floor(a(n)^3 / 10^A055642(a(n))), where A055642(n) is the number of digits in the decimal expansion of n. As well, let na and nb denote the indices of the preceding and next terms that begin with a 9. Then (1/q(n)) * (a(n)^4 - a(n)^3 - a(n)^2 + a(n)) - 2*a(n)^2 + a(n) + q(n) + 1 = a(na+nb-n)^2 - a(na+nb-n) - q(na+nb-n). - _Christopher Hohl_, Apr 08 2019
%D S. Premchaud, A class of numbers, Math. Student, 48 (1980), 293-300.
%H Eric M. Schmidt, <a href="/A033819/b033819.txt">Table of n, a(n) for n = 1..1000</a>
%H Robert Dawson, <a href="https://www.emis.de/journals/JIS/VOL21/Dawson/dawson6.html">On Some Sequences Related to Sums of Powers</a>, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TrimorphicNumber.html">Trimorphic Number</a>
%H <a href="/index/Ar#automorphic">Index entries for sequences related to automorphic numbers</a>
%e 376^3 = 53157376 which ends with 376.
%t Do[x=Floor[N[Log[10, n], 25]]+1; If[Mod[n^3, 10^x] == n, Print[n]], {n, 1, 10000}]
%t Select[Range[100000],PowerMod[#,3,10^IntegerLength[#]]==#&](* _Harvey P. Dale_, Nov 04 2011 *)
%t Select[Range[0, 10^5], 10^IntegerExponent[#^3-#, 10]>#&] (* _Jean-François Alcover_, Apr 04 2013 *)
%o (Magma) [n: n in [0..10^5] | Intseq(n^3)[1..#Intseq(n)] eq Intseq(n)]; // _Bruno Berselli_, Apr 04 2013
%Y Cf. A074194, A215558 (cubes of the terms).
%K base,nonn
%O 1,3
%A _David W. Wilson_