

A318961


One of the two successive approximations up to 2^n for 2adic integer sqrt(7). This is the 3 (mod 4) case.


3



3, 3, 11, 11, 11, 75, 75, 331, 843, 1867, 3915, 8011, 16203, 16203, 16203, 81739, 212811, 474955, 474955, 474955, 2572107, 6766411, 6766411, 23543627, 57098059, 57098059, 57098059, 57098059, 593968971, 1667710795, 1667710795, 1667710795, 1667710795, 18847579979
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OFFSET

2,1


COMMENTS

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).
The 2adic integers are very different from padic 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 padic integers is gcd(n, p1) 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, p1) > 1 and gcd(a, p) = 1, then the solutions to x^n = a differ start from the rightmost digit for odd primes p, while for p = 2 they differ start from the second rightmost digit. As a result, the formulas and the program here are different from those in other entries related to padic integers.


LINKS

Jianing Song, Table of n, a(n) for n = 2..999 (offset corrected by Jianing Song)
G. P. Michon, Introduction to padic integers, Numericana.


FORMULA

a(2) = 3; for n >= 3, a(n) = a(n1) if a(n1)^2 + 7 is divisible by 2^(n+1), otherwise a(n1) + 2^(n1).
a(n) = 2^n  A318960(n).
a(n) = Sum_{i=0..n1} A318963(i)*2^i.


EXAMPLE

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.
a(2)^2 + 7 = 16 which is divisible by 16, so a(3) = a(2) = 3.
a(3)^2 + 7 = 16 which is not divisible by 32, so a(4) = a(3) + 2^3 = 11.
a(4)^2 + 7 = 128 which is divisible by 64, so a(5) = a(4) = 11.
a(5)^2 + 7 = 128 which is divisible by 128, so a(6) = a(5) = 11.
...


PROG

(PARI) a(n) = if(n==2, 3, truncate(sqrt(7+O(2^(n+1)))))


CROSSREFS

Cf. A318963.
Expansions of padic integers:
A318960, this sequence (2adic, sqrt(7));
A268924, A271222 (3adic, sqrt(2));
A268922, A269590 (5adic, sqrt(4));
A048898, A048899 (5adic, sqrt(1));
A290567 (5adic, 2^(1/3));
A290568 (5adic, 3^(1/3));
A290800, A290802 (7adic, sqrt(6));
A290806, A290809 (7adic, sqrt(5));
A290803, A290804 (7adic, sqrt(3));
A210852, A212153 (7adic, (1+sqrt(3))/2);
A290557, A290559 (7adic, sqrt(2));
A286840, A286841 (13adic, sqrt(1));
A286877, A286878 (17adic, sqrt(1)).
Also expansions of 10adic integers:
A007185, A010690 (nontrivial roots to x^2x);
A216092, A216093, A224473, A224474 (nontrivial roots to x^3x).
Sequence in context: A095019 A321082 A167428 * A309798 A068594 A147175
Adjacent sequences: A318958 A318959 A318960 * A318962 A318963 A318964


KEYWORD

nonn


AUTHOR

Jianing Song, Sep 06 2018


EXTENSIONS

Offset corrected by Jianing Song, Aug 28 2019


STATUS

approved



