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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

Original entry on oeis.org

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
Offset: 1

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Author

Jianing Song, Jul 08 2019

Keywords

Examples

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

Crossrefs

Cf. A001353, A232624, A306825.
Cf. also A023022, A008683.

Programs

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

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 > 1, a(n)*a(2n) = Product_{k|n} A001353(n/k)^mu(k) = A306825(n). [Corrected by Jianing Song, Oct 31 2024]
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.

A309526 a(n) is the greatest divisor of A001353(n) that is coprime to A001353(m) for all positive integers m < n.

Original entry on oeis.org

1, 4, 15, 7, 209, 13, 2911, 97, 901, 181, 564719, 193, 7865521, 2521, 6989, 18817, 1525870529, 2701, 21252634831, 37441, 6779137, 489061, 4122901604639, 37633, 274758906449, 6811741, 6575588101, 1037623, 11140078609864049, 40321, 155161278879431551
Offset: 1

Views

Author

Jianing Song, Aug 06 2019

Keywords

Comments

Analog of A178763 and A308949.
Let b(n) = A309040(n)*gcd(A309040(n),n), then for n > 3: a(n) = b(2n) for even n and b(n)*b(2n) for odd n. It seems highly impossible that b(n) = 1 holds for n > 3, so it seems that only even-indexed terms can be primes.

Examples

			A001353(6) = 780 = 2^2 * 3 * 5 * 13. We have 2 divides A001353(2) = 2 and 3, 5 divides A001353(3) = 15, but A001353(m) is coprime to 13 for all 1 <= m < 6, so a(6) = 13.

Crossrefs

Cf. A001353, A306825, A309040.
Cf. A178763, A308949, A309525.

Programs

  • PARI
    T(n) = ([4, -1; 1, 0]^n)[2, 1]
b(n) = my(v=divisors(n)); prod(i=1, #v, T(v[i])^moebius(n/v[i]))
a(n) = if(isprime(n)&&!(12%n), b(n), b(n)/gcd(n, b(n)))

Formula

a(n) = A306825(n) / gcd(A306825(n), n) if n != 2, 3.
Showing 1-2 of 2 results.

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