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 A375090 #17 Aug 02 2024 19:02:10 %S A375090 0,1,19,3763,141631795,200595654682274611, %T A375090 402386166774103602282563565610210099, %U A375090 1619146272111567178177755907012051366495859012549915851432930874097578803 %N A375090 Omit the trailing 4 from the terms of the Lucas-Lehmer sequence A003010. %C A375090 The next term a(8) has 146 digits. %C A375090 Proposition: A003010(n) has only a trailing 4. %C A375090 Proof: the case n = 0 is trivial since a(0) = 4 has only a digit. For n > 0, we need to prove that the terms have at least a trailing 4 and that this is unique. A003010(n) == 4 (mod 10) because A003010(0) = 4 and 4^2 - 2 = 14 == 4 (mod 10). To prove that this is unique, we prove that the tenth digit of A003010(n) is never equal to 4 for n > 0, or equivalently, that it is odd. Considering that (10*h + 4)^2 = 100*h^2 + 80*h + 16 and 80*h + 16 has the tenth digit of the form 8*k + 1 mod 10, it follows that the tenth digit of A003010(n-1)^2 is odd, and therefore, also that of A003010(n-1)^2 - 2 = A003010(n). QED. %H A375090 Stefano Spezia, <a href="/A375090/b375090.txt">Table of n, a(n) for n = 0..10</a> %F A375090 a(n) = (A003010(n) - 4)/10. %F A375090 a(n) = 10*a(n-1)^2 + 8*a(n-1) + 1 for n > 0. %t A375090 A003010[0]=4; A003010[n_]:=A003010[n-1]^2-2; Array[A003010, 9, 0]; a[n_]:=(A003010[n]-4)/10; Array[a,8,0] %Y A375090 Cf. A003010, A017077. %K A375090 nonn,base %O A375090 0,3 %A A375090 _Stefano Spezia_, Jul 29 2024