This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A307600 #31 Jan 18 2020 15:31:20 %S A307600 0,1,21,463,464,9999,10000,215443,4641588,99999999,100000000, %T A307600 2154434689,2154434690,46415888335,46415888336,999999999999, %U A307600 1000000000000,21544346900318,464158883361277,9999999999999999,10000000000000000,215443469003188371,215443469003188372 %N A307600 Numbers k such that the digits of k^(1/4) begin with k. %C A307600 Program is in A307371. %C A307600 From _Bernard Schott_, May 01 2019: (Start) %C A307600 There are two nontrivial families in this sequence: %C A307600 1st: 21, 215443, 2154434689, 2154434690, 21544346900318, ... %C A307600 2nd: 463, 464, 4641588, 46415888335, 46415888336, ... (End) %C A307600 From _Jon E. Schoenfield_, May 04 2019: (Start) %C A307600 For each number k such that the digits of k^(1/m) begin with k, we have, for each m >= 2, floor(k^(1/m) * 10^d) = k for some integer d, so k^(1/m) * 10^d ~= k; solving for k gives k ~= 10^(d*m/(m-1)). %C A307600 In the m=4 case (this sequence), this gives k ~= 10^(d*4/3) so, as d is incremented by 1, 10^(d*4/3) increases by a factor of 10^(4/3) = 10000^1/3 = 21.5443469...: %C A307600 . %C A307600 d | 10^(d*4/3) %C A307600 ---+--------------------- %C A307600 0 | 1 %C A307600 1 | 21.544... %C A307600 2 | 464.158... %C A307600 3 | 10000 %C A307600 4 | 215443.469... %C A307600 5 | 4641588.833... %C A307600 6 | 100000000 %C A307600 7 | 2154434690.031... %C A307600 8 | 46415888336.127... %C A307600 9 | 1000000000000 %C A307600 . %C A307600 Each nonnegative integer d corresponds to one or two terms in the sequence. Letting j = floor(10000^(d/3)), j is necessarily a term; j-1 is also a term iff (j-1)^(1/4)*10^d < j. This inequality is satisified %C A307600 for d == 1 (mod 3) at d = 7, 13, 16, 34, 37, ...; %C A307600 for d == 2 (mod 3) at d = 2, 8, 20, 29, 32, 35, ...; %C A307600 and at every d == 0 (mod 3). %C A307600 (The sequence contains no other terms than numbers k of the form j or j-1 where j = floor(10000^(d/3)) for some nonnegative integer d.) %C A307600 (End) %H A307600 Chai Wah Wu, <a href="/A307600/b307600.txt">Table of n, a(n) for n = 1..1172</a> %e A307600 215443^(1/4) = 21.544335..., which begins with "215443", so 215443 is in the sequence. %Y A307600 Cf. A307371, A307588. %Y A307600 Cf. A052211 (analog for 4th power instead of 1/4). %K A307600 nonn,base %O A307600 1,3 %A A307600 _Dmitry Kamenetsky_, Apr 17 2019 %E A307600 a(12)-a(23) from _Jon E. Schoenfield_, May 01 2019