A117318 -id:A117318 - cp's OEIS frontend

A117223 Numbers n such that Phi(n,x) is a flat cyclotomic polynomial of order three.

231, 399, 435, 465, 483, 651, 663, 741, 861, 885, 903, 915, 1113, 1173, 1209, 1281, 1311, 1335, 1353, 1443, 1479, 1533, 1581, 1599, 1653, 1743, 1833, 1947, 2163, 2211, 2235, 2247, 2265, 2301, 2337, 2379, 2409, 2485, 2667, 2685, 2715, 2829, 2877, 2915

Offset: 1

T. D. Noe, Mar 04 2006

nonn

A flat polynomial is defined to be a polynomial whose coefficients are -1, 0, or 1. Order three means that n is the product of three odd primes p < q < r. Bachman shows that for each p there are an infinite number of pairs {q,r} that generate flat cyclotomic polynomials. It is well known that all cyclotomic polynomials of orders one and two are flat. There are no flat cyclotomic polynomials of order four for n < 10^5.

Kaplan shows that the sequence also includes pqr if r = +-1 (mod pq). Sequence A160353 lists the subsequence of all odd numbers of this form, while A160355 lists the elements which are not of this form. More cases are covered by David Broadhurst's conjectures, cf. link. - M. F. Hasler, May 15 2009

David Broadhurst and T. D. Noe, Table of n, a(n) for n = 1..10000
Gennady Bachman, Flat cyclotomic polynomials of order three, Bull. London Math. Soc. 38 (2006), 53-60.
David Broadhurst, Flat ternary cyclotomic polynomials, in: Yahoo! group "primenumbers", May 2009. [Broken link]
Phil Carmody and others, Cyclotomic polynomial puzzles, digest of 43 messages in primenumbers Yahoo group, May 9 - May 23, 2009.
Nathan Kaplan, Flat cyclotomic polynomials of order three, J. Number Theory 127 (2007), 118-126.
Carlo Sanna, A Survey on Coefficients of Cyclotomic Polynomials, arXiv:2111.04034 [math.NT], 2021.

Cf. A117318 (fourth-order flat cyclotomic polynomials).

IsOrder3[n_] := (n>1) && OddQ[n] && Transpose[FactorInteger[n]][[2]] == {1,1,1}; PolyHeight[p_] := Max[Abs[CoefficientList[p,x]]]; Clear[x]; Select[Range[4000], IsOrder3[ # ] && PolyHeight[Cyclotomic[ #,x]]==1&]

A117223(n,show=0)={ my(pqr=1,f); while(n, matsize(f=factor(pqr+=2))[1]==3 & vecmax(f[,2])==1 & vecmax(abs(Vec(polcyclo(pqr))))==1 & n-- & show & print1(pqr", ")); pqr } \\ M. F. Hasler, May 15 2009

Equals A160353 union A160355 = A160350 \ A075819 = A160350 intersect A046389. - M. F. Hasler, May 15 2009

A013590 Numbers k such that Phi(k,x) is a cyclotomic polynomial containing a coefficient with an absolute value greater than one.

Original entry on oeis.org

105, 165, 195, 210, 255, 273, 285, 315, 330, 345, 357, 385, 390, 420, 429, 455, 495, 510, 525, 546, 555, 561, 570, 585, 595, 609, 615, 627, 630, 645, 660, 665, 690, 705, 714, 715, 735, 759, 765, 770, 777, 780, 795, 805, 819, 825, 840, 855

Offset: 1

Peter T. Wang (peterw(AT)cco.caltech.edu)

nonn

Previous name was: Orders of cyclotomic polynomials containing a coefficient with an absolute value greater than one.
First occurrence of A137979(n)=k is given in A013594.
From David A. Corneth, Apr 21 2018: (Start)
Terms are composite.
If k is a term of the sequence then so is k * m for m > 0.
Let a primitive term p of this sequence be a term of which no divisor is in the sequence. Then p is an odd squarefree number. (End)

Robert G. Wilson v, Table of n, a(n) for n = 1..1627

Cf. A013594, A117223, A117318, A137979.

Flat cyclotomic polynomial: A117223 (order 3), A117318 (order 4).

isA013590 := proc(n)
    numtheory[cyclotomic](n,x) ;
    {coeffs(%,x)} ;
    map(abs,%) ;
    if % minus {1}  = {} then
        false;
    else
        true;
    end if;
end proc:
for n from 1 do
    if isA013590(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Nov 28 2016

S[ n_ ] := For[ j=1; t=0, j1 ]; If[ Length[ t ]!=0, Print[ j ] ] ]; S[ 856 ]
f[n_] := Max@ Abs@ CoefficientList[ Cyclotomic[n, x], x]; Select[ Range@ 1000, f@# > 1 &] (* Robert G. Wilson v *)
Select[Range[900],Max[Abs[CoefficientList[Cyclotomic[#,x],x]]]>1&] (* Harvey P. Dale, Mar 13 2013 *)

is(n)=for(k=0,n,if(abs(polcoeff(polcyclo(n),k))>1,return(n)));0
for(n=1,1000,if(is(n),print1(n,", "))) \\ Derek Orr, Apr 22 2015

Definition clarified by Harvey P. Dale, Mar 13 2013

New name from Michel Marcus, Apr 29 2018

A152955 Odd squarefree numbers n such that the cyclotomic polynomial Phi(n,x) has height 1.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 29, 31, 33, 35, 37, 39, 41, 43, 47, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 107, 109, 111, 113, 115, 119, 123, 127, 129, 131, 133, 137, 139, 141, 143, 145, 149, 151, 155, 157

Offset: 1

T. D. Noe, Dec 16 2008

nonn

The height of a polynomial is the maximum of the absolute value of its coefficients. Polynomials of height 1 are also called flat polynomials. This sequence includes prime (first order) and semiprime (second order) n, as well as third-order (A117223), fourth-order (A117318) and higher-order n.

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

Cf. A152940, A152941, A152942, A152943.

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

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

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, 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

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

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;
```

A114735 Least odd number k such that Phi(k,x) is a flat cyclotomic polynomial of order n.

Original entry on oeis.org

3, 15, 231, 431985

Offset: 1

T. D. Noe, Mar 14 2006

A flat polynomial is defined to be a polynomial whose coefficients are -1, 0, or 1. Order n means that k is the product of n distinct odd primes. Although the first four numbers are triangular (A000217), this appears to be a coincidence. Are there flat cyclotomic polynomials of all orders?
Conjecture that the next two terms are 746443728915 = 3 * 5 * 31 * 929 * 1727939 and 7800513423460801052132265 = 3 * 5 * 31 * 929 * 1727941 * 10450224300389. [T. D. Noe, Apr 13 2010]
In 2010, Andrew Arnold reported to me that the order of 746443728915 is 3. His paper has details about how the computation was done. - T. D. Noe, Mar 20 2013

Andrew Arnold and Michael Monagan, Calculating cyclotomic polynomials, Mathematics of Computation 80 (276) (2011) 2359-2379; preprint.

Cf. A117223 (third-order flat cyclotomic polynomials), A117318 (fourth-order flat cyclotomic polynomials).

cp's OEIS Frontend

A117223 Numbers n such that Phi(n,x) is a flat cyclotomic polynomial of order three.

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A013590 Numbers k such that Phi(k,x) is a cyclotomic polynomial containing a coefficient with an absolute value greater than one.

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A152955 Odd squarefree numbers n such that the cyclotomic polynomial Phi(n,x) has height 1.

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

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

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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.

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A114735 Least odd number k such that Phi(k,x) is a flat cyclotomic polynomial of order n.

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