A154430 -id:A154430 - cp's OEIS frontend

A152940 Odd squarefree numbers n such that the cyclotomic polynomial Phi(n,x) has height 2.

105, 165, 195, 255, 273, 285, 345, 357, 429, 455, 555, 561, 609, 615, 627, 645, 705, 715, 759, 777, 795, 805, 897, 957, 969, 987, 1001, 1005, 1015, 1023, 1045, 1065, 1085, 1095, 1105, 1131, 1185, 1221, 1239, 1245, 1265, 1295, 1407, 1419, 1435, 1455, 1491

Offset: 1

T. D. Noe, Dec 16 2008, Jan 09 2009

nonn

The height of a polynomial is the maximum of the absolute value of its coefficients. Subsequence of A154430.

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

Cf. A152955, A152941, A152942, A152943.

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

A152941 Odd squarefree numbers n such that the cyclotomic polynomial Phi(n,x) has height 3.

Original entry on oeis.org

385, 595, 665, 935, 1155, 1235, 1309, 1463, 1495, 1729, 1855, 1955, 2065, 2261, 2465, 2665, 2795, 2821, 3045, 3059, 3115, 3145, 3289, 3451, 3619, 3655, 3857, 4123, 4277, 4389, 4403, 4433, 4669, 4697, 4745, 4807, 4879, 4935, 4945, 5015, 5083, 5117, 5285

Offset: 1

T. D. Noe, Dec 16 2008, Jan 09 2008

nonn

The height of a polynomial is the maximum of the absolute value of its coefficients. Subsequence of A154430.

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

Cf. A152955, A152940, A152942, A152943.

is(n)=issquarefree(n) && n%2 && vecmax(abs(Vec(polcyclo(n))))==3 \\ Charles R Greathouse IV, Nov 05 2017

A152942 Odd squarefree numbers n such that the cyclotomic polynomial Phi(n,x) has height 4.

Original entry on oeis.org

1365, 1995, 2415, 2431, 2737, 2849, 3003, 3255, 3315, 3553, 3689, 4081, 4147, 4199, 4305, 4485, 4515, 4543, 4991, 5593, 5681, 5865, 6045, 6105, 6251, 6405, 6409, 6555, 6721, 6851, 7049, 7395, 7469, 7665, 7667, 7755, 7777, 7905, 8547, 8715, 8835, 9165

Offset: 1

T. D. Noe, Dec 16 2008, Jan 09 2009

nonn

The height of a polynomial is the maximum of the absolute value of its coefficients. Subsequence of A154430.

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

Cf. A152955, A152940, A152941, A152943.

is(n)=issquarefree(n) && n%2 && vecmax(abs(Vec(polcyclo(n))))==4 \\ Charles R Greathouse IV, Nov 05 2017

A152943 Odd squarefree numbers n such that the cyclotomic polynomial Phi(n,x) has height 5.

Original entry on oeis.org

1785, 2145, 2717, 3705, 3795, 3885, 3927, 4785, 4845, 5005, 5115, 5187, 5291, 5313, 5655, 6765, 7035, 7215, 7293, 8211, 8265, 8385, 8569, 8855, 9269, 9435, 9735, 9867, 10065, 10545, 10659, 10857, 10965, 11055, 11339, 11685, 12243, 12597, 12673

Offset: 1

T. D. Noe, Dec 16 2008, Jan 09 2009

nonn

The height of a polynomial is the maximum of the absolute value of its coefficients. Subsequence of A154430.

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

Cf. A152955, A152940, A152941, A152942.

is(n)=issquarefree(n) && n%2 && vecmax(abs(Vec(polcyclo(n))))==5 \\ Charles R Greathouse IV, Nov 05 2017

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

A344706 Odd squarefree 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, 1155, 1173, 1309, 1365, 1463, 1479, 1495, 1615, 1729, 1767, 1785, 1955, 1995, 2001, 2015, 2093, 2145, 2185, 2233, 2261, 2387, 2415, 2431, 2465, 2665, 2717, 2737, 2755, 2795, 2805, 2829, 2849, 3003, 3045, 3059, 3135, 3145, 3255

Offset: 1

Jianing Song, May 26 2021

nonn

Odd squarefree numbers in A344673.

Note that (i) for odd k, Phi_{2*k}(x) = Phi_k(-x); (ii) for prime p dividing k, Phi_{p*k}(x) = Phi_k(x^p). As a result, every term of A344673 can be written as 2^e * (p_1)^(e_1) * (p_2)^(e_2) * ... (p_r)^(e_r) * k, where k is a term of this sequence, p_1, p_2, ..., p_r are the distinct prime factors of k.

			665 = 5 * 7 * 19, 1/Phi_665(x) = 1 - x + x^5 - x^6 + x^7 - x^8 + x^10 - x^11 + x^12 - x^13 + x^14 - x^16 + x^17 - x^18 + 2*x^19 + ..., the coefficient of x^19 is 2, so 665 is a term.
1001 = 7 * 11 * 13, 1/Phi_1001(x) = 1 - x + x^7 - x^8 + x^11 - x^12 + x^13 - x^15 + x^18 - x^19 + x^20 - x^23 + x^24 - x^30 + x^31 + x^33 - x^34 + x^35 - x^36 + x^39 - x^41 + x^42 - x^43 + x^44 - x^45 + 2*x^46 + ..., the coefficient of x^46 is 2, so 1001 is a term.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000 (first 500 terms from Jianing Song)

Cf. A306453, A333248, A131672, A154430.

Proper subsequence of A344673.

fQ[n_] := Max@ Union@ Abs@ CoefficientList[ Simplify[(x^n - 1)/Cyclotomic[n, x]], x] > 1; Select[1 + 2Range@ 1500, SquareFreeQ@# && fQ@# &] (* Robert G. Wilson v, May 29 2021 *)

isA344706(k) = (k%2==1) && issquarefree(k) && (vecmax(abs(Vec((x^k-1)/polcyclo(k))))>=2)

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

cp's OEIS Frontend

A152940 Odd squarefree numbers n such that the cyclotomic polynomial Phi(n,x) has height 2.

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A152941 Odd squarefree numbers n such that the cyclotomic polynomial Phi(n,x) has height 3.

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A152942 Odd squarefree numbers n such that the cyclotomic polynomial Phi(n,x) has height 4.

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A152943 Odd squarefree numbers n such that the cyclotomic polynomial Phi(n,x) has height 5.

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

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A344706 Odd squarefree numbers k such that the expansion of the inverse of the k-th cyclotomic polynomial has a coefficient other than -1, 0 or 1.

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A189936 Odd numbers in A076763.

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