# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a318961 Showing 1-1 of 1 %I A318961 #44 Dec 17 2021 08:26:45 %S A318961 3,3,11,11,11,75,75,331,843,1867,3915,8011,16203,16203,16203,81739, %T A318961 212811,474955,474955,474955,2572107,6766411,6766411,23543627, %U A318961 57098059,57098059,57098059,57098059,593968971,1667710795,1667710795,1667710795,1667710795,18847579979 %N A318961 One of the two successive approximations up to 2^n for 2-adic integer sqrt(-7). This is the 3 (mod 4) case. %C A318961 a(n) is the unique number k in [1, 2^n] and congruent to 3 (mod 4) such that k^2 + 7 is divisible by 2^(n+1). %C A318961 The 2-adic integers are very different from p-adic ones where p is an odd prime. For example, provided that there is at least one solution, the number of solutions to x^n = a over p-adic integers is gcd(n, p-1) for odd primes p and gcd(n, 2) for p = 2. For odd primes p, x^2 = a is solvable iff a is a quadratic residue modulo p, while for p = 2 it's solvable iff a == 1 (mod 8). If gcd(n, p-1) > 1 and gcd(a, p) = 1, then the solutions to x^n = a differ starting at the rightmost digit for odd primes p, while for p = 2 they differ starting at the next-to-rightmost digit. As a result, the formulas and the program here are different from those in other entries related to p-adic integers. %H A318961 Jianing Song, Table of n, a(n) for n = 2..999 (offset corrected by Jianing Song) %H A318961 G. P. Michon, Introduction to p-adic integers, Numericana. %F A318961 a(2) = 3; for n >= 3, a(n) = a(n-1) if a(n-1)^2 + 7 is divisible by 2^(n+1), otherwise a(n-1) + 2^(n-1). %F A318961 a(n) = 2^n - A318960(n). %F A318961 a(n) = Sum_{i=0..n-1} A318963(i)*2^i. %e A318961 The unique number k in [1, 4] and congruent to 3 modulo 4 such that k^2 + 7 is divisible by 8 is 3, so a(2) = 3. %e A318961 a(2)^2 + 7 = 16 which is divisible by 16, so a(3) = a(2) = 3. %e A318961 a(3)^2 + 7 = 16 which is not divisible by 32, so a(4) = a(3) + 2^3 = 11. %e A318961 a(4)^2 + 7 = 128 which is divisible by 64, so a(5) = a(4) = 11. %e A318961 a(5)^2 + 7 = 128 which is divisible by 128, so a(6) = a(5) = 11. %e A318961 ... %o A318961 (PARI) a(n) = if(n==2, 3, truncate(sqrt(-7+O(2^(n+1))))) %Y A318961 Cf. A318963. %Y A318961 Expansions of p-adic integers: %Y A318961 A318960, this sequence (2-adic, sqrt(-7)); %Y A318961 A268924, A271222 (3-adic, sqrt(-2)); %Y A318961 A268922, A269590 (5-adic, sqrt(-4)); %Y A318961 A048898, A048899 (5-adic, sqrt(-1)); %Y A318961 A290567 (5-adic, 2^(1/3)); %Y A318961 A290568 (5-adic, 3^(1/3)); %Y A318961 A290800, A290802 (7-adic, sqrt(-6)); %Y A318961 A290806, A290809 (7-adic, sqrt(-5)); %Y A318961 A290803, A290804 (7-adic, sqrt(-3)); %Y A318961 A210852, A212153 (7-adic, (1+sqrt(-3))/2); %Y A318961 A290557, A290559 (7-adic, sqrt(2)); %Y A318961 A286840, A286841 (13-adic, sqrt(-1)); %Y A318961 A286877, A286878 (17-adic, sqrt(-1)). %Y A318961 Also expansions of 10-adic integers: %Y A318961 A007185, A010690 (nontrivial roots to x^2-x); %Y A318961 A216092, A216093, A224473, A224474 (nontrivial roots to x^3-x). %K A318961 nonn %O A318961 2,1 %A A318961 _Jianing Song_, Sep 06 2018 %E A318961 Offset corrected by _Jianing Song_, Aug 28 2019 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE