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A309040 a(n) = MPR2(n, 4), where MPR2(n, x) is the (monic) minimal polynomial of 2*cos(2*Pi/n) as defined in A232624. 1
2, 6, 5, 4, 19, 3, 71, 14, 53, 11, 989, 13, 3691, 41, 145, 194, 51409, 51, 191861, 181, 2017, 571, 2672279, 193, 524899, 2131, 140453, 2521, 138907099, 241, 518408351, 37634, 391249, 29681, 5352481, 2701, 26947261171, 110771, 5449393, 37441, 375326930089 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Table of n, a(n) for n=1..41.

FORMULA

By the comment in A232624, we have: A001353(n) = Product_{k|2n, k>=3} MPR2(k, 4) = Product_{k|2n, k>=3} a(k).

a(n) = Product_{0<=m<=n/2, gcd(m, n)=1} (4 - 2*cos(2Pi*m/n)).

If 4 divides n, then a(n) = Product_{k|(n/2)} A001353((n/2)/k)^mu(k) = A306825(n/2), where mu = A008683. For odd n, a(n)*a(2n) = Product_{k|n} A001353(n/k)^mu(k) = A306825(n).

Let b(n) = MPR2(n, -4)*(-1)^A023022(n) for n > 2, then a(n) = b(2n) for odd n, a(n) = b(n/2) for n congruent to 4 modulo 2, a(n) = b(n) for n divisible by 4.

EXAMPLE

MPR2(15, x) = x^4 - x^3 - 4x^2 + 4x + 1, so a(15) = MPR2(15, 4) = 145.

MATHEMATICA

a[n_] := (p = MinimalPolynomial[2*Cos[2*(Pi/n)], 4]; p); Table[a[n], {n, 1, 40}]

CROSSREFS

Cf. A001353, A232624, A306825.

Cf. also A023022, A008683.

Sequence in context: A198821 A171897 A105029 * A316134 A273621 A190124

Adjacent sequences:  A309037 A309038 A309039 * A309041 A309042 A309043

KEYWORD

nonn

AUTHOR

Jianing Song, Jul 08 2019

STATUS

approved

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Last modified December 7 20:19 EST 2021. Contains 349588 sequences. (Running on oeis4.)