A160340 Indices of records in heights of cyclotomic polynomials (A160338).
1, 105, 385, 1365, 1785, 2805, 3135, 6545, 10465, 11305, 17255, 20615, 26565, 40755, 106743, 171717, 255255, 279565, 327845, 707455, 886445, 983535, 1181895, 1752465, 3949491, 8070699, 10163195, 13441645, 15069565, 30489585, 37495115, 40324935
Offset: 1
Keywords
A160339 Records in heights of cyclotomic polynomials (A160338).
1, 2, 3, 4, 5, 6, 7, 9, 14, 23, 25, 27, 59, 359, 397, 434, 532, 1182, 31010, 35111, 44125, 59815, 14102773, 14703509, 56938657, 74989473, 1376877780831, 1475674234751, 1666495909761, 2201904353336, 2286541988726, 2699208408726, 862550638890874931
Offset: 1
Keywords
Links
- Max Alekseyev, Table of n, a(n) for n = 1..42
- A. Arnold and M. Monagan, Calculating cyclotomic polynomials of very large height.
A137979 Highest coefficient occurring in the factorization of x^n - 1 over the reals.
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2
Offset: 1
Keywords
Comments
Based on a comment in Mathematica helpfile ref/Factor - Neat Examples.
The first factorization of x^n - 1 in which a 2 appears as a coefficient is for n=105.
Different from A160338, see comment there.
Examples
a(4) = 1 because x^4 - 1 = (x^2+1)(x+1)(x-1) and the highest coefficient of these three terms is 1. The first time a 2 appears is at n=105, where the factorization is: (x-1)*(x^6+x^5+x^4+x^3+x^2+x+1)*(x^4+x^3+x^2+x+1)* (x^24-x^23+x^19-x^18+x^17-x^16+x^14-x^13+x^12-x^11+x^10-x^8+x^7-x^6+x^5-x+1)* (x^2+x+1)*(x^12-x^11+x^9-x^8+x^6-x^4+x^3-x+1)* (x^8-x^7+x^5-x^4+x^3-x+1)* (x^48+x^47+x^46-x^43-x^42-2*x^41-x^40-x^39+x^36+x^35+x^34+x^33+x^32+x^31-x^28-x^26-x^24-x^22-x^20+x^17+x^16+x^15+x^14+x^13+x^12-x^9-x^8-2*x^7-x^6-x^5+x^2+x+1). - _N. J. A. Sloane_, Apr 18 2008
Links
- N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Programs
-
Mathematica
Table[Max[Abs[Flatten[CoefficientList[Transpose[FactorList[x^i - 1]][[1]], x]]]], {i, 1, 1000}] -
PARI
a(n) = {my(f = factor(x^n-1)); vecmax(vector(#f~, k, vecmax(apply(x->abs(x), Vec(f[k,1])))));} \\ Michel Marcus, Dec 05 2018
A365335 The number of exponentially odd coreful divisors of the square root of the largest square dividing n.
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1
Comments
The number of divisors of the square root of the largest square dividing n is A046951(n).
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
-
Mathematica
f[p_, e_] := Max[1, Floor[(e+2)/4]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
-
PARI
a(n) = vecprod(apply(x -> max(1, (x+2)\4), factor(n)[, 2]));
Formula
a(n) > 1 if and only if n is a bicubeful number (A355265).
Multiplicative with a(p^e) = floor((e+2)/4).
Dirichlet g.f.: zeta(s) * zeta(4*s) * Product_{p prime} (1 - 1/p^(4*s) + 1/p^(6*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(4) * Product_{p prime} (1 - 1/p^4 + 1/p^6) = 1.0181534831085... .
Extensions
Name corrected by Amiram Eldar, Sep 08 2023
A160341 Height of the 3*5*7*11*...*prime(n)-th cyclotomic polynomial.
1, 1, 2, 3, 23, 532, 669606, 8161018310, 2888582082500892851
Offset: 1
Links
- A. Arnold and M. Monagan, Calculating cyclotomic polynomials of very large height.
A189408 Least k where Phi(k) has height greater than k^n, where Phi(k) is the k-th cyclotomic polynomial and the height is the largest absolute value of the coefficients.
1181895, 43730115, 416690995, 1880394945
Offset: 1
Comments
Arnold & Monagan compute this sequence to demonstrate their fast algorithm for computing cyclotomic polynomials.
This sequence is infinite because (the supremum of) A160338 grows exponentially.
Links
- Andrew Arnold and Michael Monagan, A high-performance algorithm for calculating cyclotomic polynomials, PASCO 2010. doi:10.1145/1837210.1837228
- Andrew Arnold and Michael Monagan, A fast recursive algorithm for computing cyclotomic polynomials, ACM Commun. Comput. Algebra 44:3/4 (2010), pp. 89-90. doi:10.1145/1940475.1940479
- Andrew Arnold and Michael Monagan, Calculating cyclotomic polynomials, Mathematics of Computation 80 (276) (2011) 2359-2379 preprint.
- Andrew Arnold and Michael Monagan, Cyclotomic Polynomials
Comments
Links
Crossrefs
Programs
Mathematica
r = 0; Do[If[# > r, r = #; Print[n]] &@ Max@ Abs@ CoefficientList[Cyclotomic[n, x], x], {n, 10^4}] (* Michael De Vlieger, May 20 2024 *)PARI
print1(r=1); for(n=2,1e4, t=vecmax(abs(Vec(polcyclo(n)))); if(t>r, r=t; print1(", "n))) \\ Charles R Greathouse IV, Jun 28 2012