A290800 One of the two successive approximations up to 7^n for the 7-adic integer sqrt(-6). These are the numbers congruent to 1 mod 7 (except for the initial 0).

0, 1, 22, 120, 120, 9724, 26531, 144180, 144180, 17438583, 259560225, 259560225, 259560225, 83307283431, 180196293838, 2893088585234, 17135773115063, 116834564823866, 582095592798280, 10352577180260974, 55948157921753546, 454909489409813551

Offset: 0

Author

Seiichi Manyama, Aug 10 2017

Keywords

• nonn

Comments

x = ...140231,

x^2 = ...666661 = -6.

Examples

			a(1) =     1_7 = 1,
a(2) =    31_7 = 22,
a(3) =   231_7 = 120,
a(4) =   231_7 = 120,
a(5) = 40231_7 = 9724.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1183
• Wikipedia, Hensel's Lemma.

Crossrefs

Cf. A290794, A290802.

Programs

Maple

with(padic):
R:= [rootp(x^2+6,7,100)]:
R1:= op(select(t -> ratvaluep(evalp(t,7,1))=1, R)):
seq(ratvaluep(evalp(R1,7,n)),n=0..100); # Robert Israel, Aug 11 2017

PARI

a(n) = if (n, truncate(sqrt(-6+O(7^(n)))), 0)

Formula

a(0) = 0 and a(1) = 1, a(n) = a(n-1) + 3 * (a(n-1)^2 + 6) mod 7^n for n > 1.