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 A325893 #10 Sep 08 2019 02:24:31 %S A325893 0,1,1,5,5,21,21,21,149,149,149,1173,1173,1173,9365,9365,42133,107669, %T A325893 238741,500885,1025173,1025173,1025173,1025173,1025173,1025173, %U A325893 34579605,34579605,34579605,34579605,571450517,1645192341,1645192341,1645192341,10235126933,10235126933 %N A325893 The successive approximations up to 2^n for 2-adic integer 5^(1/5). %C A325893 a(n) is the unique solution to x^5 == 5 (mod 2^n) in the range [0, 2^n - 1]. %H A325893 Wikipedia, <a href="https://en.wikipedia.org/wiki/P-adic_number">p-adic number</a> %F A325893 For n > 0, a(n) = a(n-1) if a(n-1)^5 - 5 is divisible by 2^n, otherwise a(n-1) + 2^(n-1). %e A325893 For n = 2, the unique solution to x^5 == 5 (mod 4) in the range [0, 3] is x = 1, so a(2) = 1. %e A325893 a(2)^5 - 5 = -4 which is not divisible by 8, so a(3) = a(2) + 4 = 5; %e A325893 a(3)^5 - 5 = 3120 which is divisible by 16, so a(4) = a(3) = 5; %e A325893 a(4)^5 - 5 = 3120 which is not divisible by 32, so a(5) = a(4) + 16 = 21; %e A325893 a(5)^5 - 5 = 4084096 which is divisible by 64, so a(6) = a(5) = 21. %o A325893 (PARI) a(n) = lift(sqrtn(5+O(2^n), 5)) %Y A325893 For the digits of 5^(1/5), see A325897. %Y A325893 Approximations of p-adic fifth-power roots: %Y A325893 A325892 (2-adic, 3^(1/5)); %Y A325893 this sequence (2-adic, 5^(1/5)); %Y A325893 A325894 (2-adic, 7^(1/5)); %Y A325893 A325895 (2-adic, 9^(1/5)); %Y A325893 A322157 (5-adic, 7^(1/5)); %Y A325893 A309450 (7-adic, 2^(1/5)); %Y A325893 A309451 (7-adic, 3^(1/5)); %Y A325893 A309452 (7-adic, 4^(1/5)); %Y A325893 A309453 (7-adic, 5^(1/5)); %Y A325893 A309454 (7-adic, 6^(1/5)). %K A325893 nonn %O A325893 0,4 %A A325893 _Jianing Song_, Sep 07 2019