A341538 - OEIS

A341538

One of the two successive approximations up to 2^n for 2-adic integer sqrt(17). This is the 1 (mod 4) case.

1, 1, 9, 9, 41, 105, 233, 233, 745, 1769, 1769, 1769, 9961, 9961, 9961, 75497, 206569, 206569, 206569, 1255145, 1255145, 5449449, 13838057, 30615273, 64169705, 64169705, 64169705, 332605161, 869476073, 869476073, 869476073, 5164443369, 13754377961, 13754377961

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OFFSET

2,3

COMMENTS

a(n) is the unique number k in [1, 2^n] and congruent to 1 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) = 1; 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 - A341539(n).

a(n) = Sum_{i=0..n-1} A322217(i)*2^i.

EXAMPLE

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

a(2)^2 - 17 = -16 which is divisible by 16, so a(3) = a(2) = 1.

a(3)^2 - 17 = -16 which is not divisible by 32, so a(4) = a(3) + 2^3 = 9.

a(4)^2 - 17 = 64 which is divisible by 64, so a(5) = a(4) = 9.

a(5)^2 - 17 = 64 which is not divisible by 128, so a(6) = a(5) + 2^5 = 41.

...

PROG

(PARI) a(n) = truncate(sqrt(17+O(2^(n+1))))

CROSSREFS

Cf. A341539 (the 3 (mod 4) case), A322217 (digits of the associated 2-adic square root of 17), A318960, A318961 (successive approximations of sqrt(-7)).

Sequence in context: A097988 A103646 A246314 * A325895 A111219 A339341

Adjacent sequences: A341535 A341536 A341537 * A341539 A341540 A341541

KEYWORD

nonn

AUTHOR

Jianing Song, Feb 13 2021

STATUS

approved