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A341539 One of the two successive approximations up to 2^n for 2-adic integer sqrt(17). This is the 3 (mod 4) case. 5
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
Jianing Song, Feb 13 2021
STATUS
approved

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Last modified March 28 04:13 EDT 2024. Contains 371235 sequences. (Running on oeis4.)