A327302 One of the two successive approximations up to 5^n for the 5-adic integer sqrt(-9). This is the 1 (mod 5) case (except for n = 0).

0, 1, 21, 46, 546, 3046, 12421, 59296, 59296, 840546, 8653046, 28184296, 125840546, 369981171, 2811387421, 2811387421, 2811387421, 460575059296, 2749393418671, 10378787949921, 48525760606171, 143893192246796, 2051241825059296, 6819613407090546, 30661471317246796

Offset

0,3

Comments

a(n) is the unique number k in [1, 5^n] and congruent to 1 mod 5 such that k^2 + 9 is divisible by 5^n.

Links

Table of n, a(n) for n=0..24.

G. P. Michon, Introduction to p-adic integers, Numericana.

Formula

a(1) = 1; for n >= 2, a(n) is the unique number k in {a(n-1) + m*5^(n-1) : m = 0, 1, 2, 3, 4} such that k^2 + 9 is divisible by 5^n.

For n > 0, a(n) = 5^n - A327303(n).

Example

The unique number k in {1, 6, 11, 16, 21} such that k^2 + 9 is divisible by 25 is k = 21, so a(2) = 21.
The unique number k in {21, 46, 71, 96, 121} such that k^2 + 9 is divisible by 125 is k = 46, so a(3) = 46.
The unique number k in {46, 171, 296, 421, 546} such that k^2 + 9 is divisible by 625 is k = 546, so a(4) = 546.

Prog

(PARI) a(n) = truncate(sqrt(-9+O(5^n)))

Crossrefs

For the digits of sqrt(-9) see A327304 and A327305.

Approximations of 5-adic square roots:

this sequence, A327303 (sqrt(-9));

A324027, A324028 (sqrt(-6));

A268922, A269590 (sqrt(-4));

A048898, A048899 (sqrt(-1));

A324023, A324024 (sqrt(6)).

Sequence in context: A044479 A139581 A250777 * A156966 A146705 A146713

Adjacent sequences: A327299 A327300 A327301 * A327303 A327304 A327305

Keyword

nonn

Author

Jianing Song, Sep 16 2019

Status

approved