A341539 - OEIS

A341539

One of the two successive approximations up to 2^n for 2-adic integer sqrt(17). This is the 3 (mod 4) case.

3, 7, 7, 23, 23, 23, 23, 279, 279, 279, 2327, 6423, 6423, 22807, 55575, 55575, 55575, 317719, 842007, 842007, 2939159, 2939159, 2939159, 2939159, 2939159, 70048023, 204265751, 204265751, 204265751, 1278007575, 3425491223, 3425491223, 3425491223, 20605360407

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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 - 17 is divisible by 2^(n+1).

LINKS

Jianing Song, Table of n, a(n) for n = 2..1000

FORMULA

a(2) = 3; for n >= 3, a(n) = a(n-1) if a(n-1)^2 - 17 is divisible by 2^(n+1), otherwise a(n-1) + 2^(n-1).

a(n) = 2^n - A341538(n).

a(n) = Sum_{i=0..n-1} A341540(i)*2^i.

EXAMPLE

The unique number k in [1, 4] and congruent to 3 modulo 4 such that k^2 - 17 is divisible by 8 is 3, so a(2) = 3.

a(2)^2 - 17 = -8 which is not divisible by 16, so a(3) = a(2) + 2^2 = 7.

a(3)^2 - 17 = 32 which is divisible by 32, so a(4) = a(3) = 7.

a(4)^2 - 17 = 32 which is not divisible by 64, so a(5) = a(4) + 2^4 = 23.

a(5)^2 - 17 = 512 which is divisible by 128, so a(6) = a(5) = 23.

...

MATHEMATICA

Table[First@Select[PowerModList[17, 1/2, 2^(k+1)], Mod[#, 4]==3&], {k, 2, 35}] (* Giorgos Kalogeropoulos, Oct 22 2022 *)

PROG

(PARI) a(n) = if(n==2, 3, truncate(-sqrt(17+O(2^(n+1)))))

CROSSREFS

Cf. A341538 (the 1 (mod 4) case), A341540 (digits of the associated 2-adic square root of 17), A318960, A318961 (successive approximations of sqrt(-7)).

Sequence in context: A270307 A261480 A121172 * A322934 A077629 A184467

Adjacent sequences: A341536 A341537 A341538 * A341540 A341541 A341542

KEYWORD

nonn

AUTHOR

Jianing Song, Feb 13 2021

STATUS

approved