The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A000188 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A000188

(1) Number of solutions to x^2 == 0 (mod n). (2) Also square root of largest square dividing n. (3) Also max_{ d divides n } gcd(d, n/d).
(history; published version)

#163 by R. J. Mathar at Sun Jan 28 08:49:18 EST 2024

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#162 by R. J. Mathar at Sun Jan 28 08:14:56 EST 2024

CROSSREFS

Cf. A240976 (Dgf at s=2).

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#161 by Jon E. Schoenfield at Thu Jan 11 00:42:02 EST 2024

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#160 by Jon E. Schoenfield at Thu Jan 11 00:41:54 EST 2024

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#159 by Jon E. Schoenfield at Thu Jan 11 00:41:31 EST 2024

FORMULA

a(2*n) = a(n)*(A096268(n-1) + 1). - observed by Velin Yanev, Jul 14 2017, The formula says that a(2n) = 2*a(n) only when 2-adic valuation of n (A007814(n)) is odd, otherwise a(2n) = a(n). This follows easily from the definition (2). - __Antti Karttunen_, Nov 28 2017

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#158 by Michael De Vlieger at Wed Oct 19 20:10:44 EDT 2022

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#157 by Kevin Ryde at Wed Oct 19 19:26:27 EDT 2022

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#156 by Kevin Ryde at Wed Oct 19 19:23:56 EDT 2022

COMMENTS

Proof that (2) = (3): Let max{[gcd(d, n/d)} = K, then d = Kx, n/d = Ky so n = KKxy where xy is the squarefree part of n, otherwise K is not maximal. Observe also that g = gcd(K, xy) is not necessarily 1. Thus K is also the "maximal square-root factor" of n. - Labos Elemer, July 2000

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Discussion

Wed Oct 19

19:24

Kevin Ryde: stray square bracket

#155 by Alois P. Heinz at Fri Aug 20 07:05:19 EDT 2021

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#154 by Ilya Gutkovskiy at Fri Aug 20 06:38:23 EDT 2021

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