A013590 -id:A013590 - cp's OEIS frontend

A154430 Odd squarefree numbers n such that the cyclotomic polynomial Phi(n,x) has height > 1.

105, 165, 195, 255, 273, 285, 345, 357, 385, 429, 455, 555, 561, 595, 609, 615, 627, 645, 665, 705, 715, 759, 777, 795, 805, 897, 935, 957, 969, 987, 1001, 1005, 1015, 1023, 1045, 1065, 1085, 1095, 1105, 1131, 1155, 1185, 1221, 1235, 1239, 1245, 1265

Offset: 1

T. D. Noe, Jan 09 2009

nonn

The height of a polynomial is the maximum of the absolute value of its coefficients. Different from A118678, which excludes terms that are a multiple of smaller terms.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A013590, A117318, A117223, A152940, A152941, A152942, A152943, A152955.

Select[Range[2000], OddQ[#] && SquareFreeQ[#] && Max[ Abs[ CoefficientList[ Cyclotomic[#, x], x]]] > 1&] (* Jean-François Alcover, Nov 14 2016 *)

is(n)=issquarefree(n) && n%2 && vecmax(abs(Vec(polcyclo(n))))>1 \\ Charles R Greathouse IV, Nov 05 2017

A137979 Highest coefficient occurring in the factorization of x^n - 1 over the reals.

Original entry on oeis.org

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

Ian Miller, Feb 25 2008

nonn

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.

			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

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A013590, A013594.

Table[Max[Abs[Flatten[CoefficientList[Transpose[FactorList[x^i - 1]][[1]], x]]]], {i, 1, 1000}]

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

A118678 Primitive orders of cyclotomic polynomials containing a coefficient with absolute value >= 2.

Original entry on oeis.org

105, 165, 195, 255, 273, 285, 345, 357, 385, 429, 455, 555, 561, 595, 609, 615, 627, 645, 665, 705, 715, 759, 777, 795, 805, 897, 935, 957, 969, 987, 1001, 1005, 1015, 1023, 1045, 1065, 1085, 1095, 1105, 1131, 1185, 1221, 1235, 1239, 1245, 1265, 1295

Offset: 1

Max Alekseyev, May 19 2006

nonn

All elements of A013590 with no proper divisors belonging to A013590.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A013590, A117318, A117223.
Cf. A152940, A152941, A152942, A152943, A152955
Cf. A154430

A318884 a(n) is the sum of absolute values of the coefficients in the n-th cyclotomic polynomial.

Original entry on oeis.org

2, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 7, 2, 17, 3, 19, 5, 9, 11, 23, 3, 5, 13, 3, 7, 29, 7, 31, 2, 15, 17, 17, 3, 37, 19, 17, 5, 41, 9, 43, 11, 7, 23, 47, 3, 7, 5, 23, 13, 53, 3, 17, 7, 25, 29, 59, 7, 61, 31, 9, 2, 31, 15, 67, 17, 31, 17, 71, 3, 73, 37, 7, 19, 31, 17, 79, 5, 3, 41, 83, 9, 41, 43, 39, 11, 89

Offset: 1

Antti Karttunen, Sep 10 2018

nonn

Differs from A051664 in the positions given by A013590, thus for the first time at n=105, where a(105) = 35, while A051664(105) = 33 as the 105th cyclotomic polynomial is the first one that has a coefficient other than 1, 0, or -1.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
Wikipedia, Cyclotomic polynomial

Cf. A013590, A013595, A051258, A051664, A318886.

Array[Total@ Abs@ CoefficientList[Cyclotomic[#, x], x] &, 89] (* Michael De Vlieger, Sep 10 2018 *)

A318884(n) = vecsum(apply(abs,Vec(polcyclo(n)))); \\ Antti Karttunen, Sep 10 2018

A344673 Numbers k such that the expansion of the inverse of the k-th cyclotomic polynomial has a coefficient other than -1, 0 or 1.

Original entry on oeis.org

561, 595, 665, 741, 935, 1001, 1105, 1122, 1155, 1173, 1190, 1309, 1330, 1365, 1463, 1479, 1482, 1495, 1615, 1683, 1729, 1767, 1785, 1870, 1955, 1995, 2001, 2002, 2015, 2093, 2145, 2185, 2210, 2223, 2233, 2244, 2261, 2310, 2346, 2380, 2387, 2415, 2431, 2465, 2618

Offset: 1

Jianing Song, May 26 2021

nonn

Define Psi_n(x) = (x^n-1)/Phi_n(x), with Phi_n(x) the n-th cyclotomic polynomial, then 1/Phi_n(x) = -Psi_n(x) * (1 + x^n + x^(2n) + ...), hence the period of coefficients in the expansion of 1/Phi_n(x) is n.
For odd k, k is a term if and only 2*k is a term.
For prime p dividing k, k is a term if and only if p*k is a term.
From Robert G. Wilson v, Jun 04 2021: (Start)
The period of the expansion of Psi(n) is A062830(n).
Terms are neither prime nor semiprime.
Least k having a run of j consecutive terms, for j >= 0: 561, 2001, 22630, 68263, ...
(End)

			1/Phi_561(x) = 1 - x + x^3 - x^4 + x^6 - x^7 + x^9 - x^10 + x^11 - x^13 + x^14 - x^16 + 2*x^17 + ..., the coefficient of x^17 is 2, so 561 is a term.
1/Phi_595(x) = 1 - x + x^5 - x^6 + x^7 - x^8 + x^10 - x^11 + x^12 - x^13 + x^14 - x^16 + 2*x^17 + ..., the coefficient of x^17 is 2, so 595 is a term.

Robert G. Wilson v, Table of n, a(n) for n = 1..11357 (first 900 terms from Jianing Song)

Cf. A306453, A333248, A131672, A013590.

A344706 is a proper subsequence.

fQ[n_] := Max@ Union@ Abs@ CoefficientList[ Simplify[(x^n - 1)/Cyclotomic[n, x]], x] > 1; Select[ Range@ 2650, fQ] (* Robert G. Wilson v, May 26 2021 *)

isA344673(k) = (vecmax(abs(Vec((x^k-1)/polcyclo(k))))>=2)

A374137 Odd numbers k such that the cyclotomic polynomial Phi(k,x) has height 1, i.e., whose coefficients are all -1, 0, or 1.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149, 151

Offset: 1

Antti Karttunen, Jul 06 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12000

Cf. A374136 (characteristic function).
Setwise difference A005408 \ A013590.
Subsequences: A152955, A117223, A117318.
Appears to be a subsequence of A089684, from which this differs for the first time at n=125, where a(125) = 257, which misses A089684(125) = 255.

PARI
```
isA374137 = A374136;
```

A189936 Odd numbers in A076763.

Original entry on oeis.org

105, 165, 195, 255, 273, 315, 345, 357, 385, 399, 465, 483, 525, 555, 585, 627, 663, 693, 705, 735, 765, 777, 795, 897, 915, 957, 975, 1005, 1095, 1113, 1155, 1173, 1185, 1281, 1295, 1305, 1353, 1365, 1515, 1545, 1575, 1617, 1677, 1683, 1725, 1755, 1785, 1815, 1935, 1953

Offset: 1

Juri-Stepan Gerasimov, May 01 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A001221, A076763, A013590, A118678, A154430.

omega := proc(n) nops( numtheory[factorset](n)) ; end proc:
isA076763 := proc(n) omega(n) > omega(n-1) and omega(n) > omega(n+1) ; end proc:
isA189936 := proc(n) type(n,'odd') and isA076763(n) ; end proc:
for n from 1 to 2000 by 2 do if isA189936(n) then printf("%d,",n) ; end if; end do;  # R. J. Mathar, May 26 2011

Select[Range[1, 2000, 2], PrimeNu[# - 1] < PrimeNu[#] > PrimeNu[# + 1]&] (* Jean-François Alcover, Nov 14 2016 *)
Select[#[[2,1]]&/@Select[Partition[Table[{n,PrimeNu[n]},{n,2000}],3,1], #[[1,2]] <#[[2,2]]>#[[3,2]]&],OddQ] (* Harvey P. Dale, Sep 15 2019 *)

a(n) = A005408(k) = A076763(m).

Corrected by R. J. Mathar, May 26 2011

A317272 Numbers k such that Phi(k,x) is a cyclotomic polynomial with middle coefficient different from -1, 0, 1.

Original entry on oeis.org

385, 770, 1155, 1365, 1540, 1925, 2065, 2145, 2310, 2415, 2465, 2695, 2717, 2730, 2737, 2821, 2849, 3003, 3080, 3255, 3315, 3465, 3619, 3850, 4095, 4130, 4199, 4235, 4277, 4290, 4543, 4620, 4785, 4830, 4845, 4879, 4930, 4991

Offset: 1

Greg Dresden, Jul 25 2018

nonn

n is in the sequence if and only if A204455(n) is in the sequence. - Robert Israel, Apr 17 2019

			385 is the first item in the list because Phi(385,x) is the first cyclotomic polynomial with middle coefficient different from -1,0,1 (the middle term is -3x^120)

Robert Israel, Table of n, a(n) for n = 1..2000
M. Beiter, The midterm coefficient of the cyclotomic polynomial F_{pq}(x), Amer. Math. Monthly 71 (1964), 769-770.
G. Dresden, On the Middle Coefficient of a Cyclotomic Polynomial, Amer. Math. Monthly 111 (2004), 531-533.

Cf. A013590, A023022 (phi(n)/2), A204455.

filter:= proc(n) local p,d;
   p:= numtheory:-cyclotomic(n,x);
   d:= degree(p,x);
   abs(coeff(p, x, d/2))>1
end proc:
select(filter, [$3..5000]); # Robert Israel, Apr 17 2019

Select[Range[3, 4000],
Abs[Coefficient[Cyclotomic[#, x], x, EulerPhi[#]/2]] > 1 &]

isok(n) = (n > 2) && (abs(polcoeff(polcyclo(n), eulerphi(n)/2)) > 1); \\ Michel Marcus, Aug 02 2018

A256575 Nonsquarefree orders of cyclotomic polynomials containing a coefficient with an absolute value greater than one.

Original entry on oeis.org

315, 420, 495, 525, 585, 630, 660, 735, 765, 780, 819, 825, 840, 855, 945, 975, 990, 1020, 1035, 1050, 1071, 1092, 1140, 1170, 1260, 1275, 1287, 1320, 1380, 1425, 1428, 1470, 1485, 1530, 1540, 1560, 1575, 1638, 1650, 1665, 1680, 1683, 1710, 1716, 1725, 1755, 1815, 1820, 1827, 1845, 1881, 1890, 1911, 1925

Offset: 1

Derek Orr, Apr 22 2015

nonn

Numbers in A013590 that are not squarefree.

Cf. A013590, A013595.

is(n)=if (!issquarefree(n), for(k=0, n, if(abs(polcoeff(polcyclo(n), k))>1, return(n))); 0)
for(n=1, 10^4, if(is(n), print1(n, ", "))) \\ edited by Michel Marcus, Apr 21 2018

A256638 Orders n of cyclotomic polynomials containing a coefficient with an absolute value greater than one, with n not divisible by 5.

Original entry on oeis.org

273, 357, 429, 546, 561, 609, 627, 714, 759, 777, 819, 858, 897, 957, 969, 987, 1001, 1023, 1071, 1092, 1122, 1131, 1218, 1221, 1239, 1254, 1287, 1309, 1407, 1419, 1428, 1463, 1491, 1518, 1547, 1551, 1554, 1638, 1659, 1677, 1683, 1716, 1729, 1749, 1767, 1771, 1794, 1827, 1869, 1881, 1887, 1911, 1914, 1938

Offset: 1

Derek Orr, Apr 22 2015

nonn

Numbers in A013590 that are not divisible by 5.

Cf. A013590, A013595.

is(n)=if(n%5, for(k=0, n, if(abs(polcoeff(polcyclo(n), k))>1, return(n))); 0)
for(n=1, 10^4, if(is(n), print1(n, ", "))) \\ edited by Michel Marcus, Apr 21 2018

cp's OEIS Frontend

A154430 Odd squarefree numbers n such that the cyclotomic polynomial Phi(n,x) has height > 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A137979 Highest coefficient occurring in the factorization of x^n - 1 over the reals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A118678 Primitive orders of cyclotomic polynomials containing a coefficient with absolute value >= 2.

Views

Author

Keywords

Comments

Links

Crossrefs

A318884 a(n) is the sum of absolute values of the coefficients in the n-th cyclotomic polynomial.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A344673 Numbers k such that the expansion of the inverse of the k-th cyclotomic polynomial has a coefficient other than -1, 0 or 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A374137 Odd numbers k such that the cyclotomic polynomial Phi(k,x) has height 1, i.e., whose coefficients are all -1, 0, or 1.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A189936 Odd numbers in A076763.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A317272 Numbers k such that Phi(k,x) is a cyclotomic polynomial with middle coefficient different from -1, 0, 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica