A117223 -id:A117223 - cp's OEIS frontend

A160355 Odd indices pqr of flat cyclotomic polynomials of order 3 which are not of the form r = +/-1 (mod pq).

231, 399, 483, 651, 663, 741, 1113, 1173, 1209, 1281, 1311, 1353, 1443, 1479, 1533, 1581, 1599, 1653, 1833, 1947, 2163, 2247, 2301, 2337, 2379, 2409, 2829, 2877, 2915, 3129, 3297, 3363, 3441, 3531, 3621, 3723, 3759, 3783, 3813, 4011, 4029, 4071, 4161

Offset: 1

M. F. Hasler, May 11 2009

nonn

This is in some sense the nontrivial part of A160350: Indeed, Kaplan (2007) has shown that Phi[pqr] has coefficients in {0,1,-1} if r = +-1 (mod pq), where pA160350 (i.e. of A117223) which do not satisfy this equality (i.e. which are not in A160353).

See A160350 for further details and references.

			a(1)=231=3*7*11 is the smallest "nontrivial" element of A160350 in the sense that it is neither of the form 2pq, and that its largest factor (11) is not congruent to +- 1 modulo the product of the smaller factors (3*7).

Robin Visser, Table of n, a(n) for n = 1..10000

Cf. A046389, A117223, A160350, A160353, A160354.

forstep( pqr=1,5999,2, my(f=factor(pqr)); #f~==3 & vecmax(f[,2])==1 & abs((f[3,1]+1)%(f[1,1]*f[2,1])-1)!=1 & vecmax(abs(Vec(polcyclo(pqr))))==1 & print1(pqr","))

Equals A117223 \ A160353 = A160354 intersect A046389.

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
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isA374137 = A374136;
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A111092 Primes congruent to {1,69} mod 70.

Original entry on oeis.org

71, 139, 211, 281, 349, 419, 421, 491, 631, 701, 769, 839, 911, 1049, 1051, 1259, 1399, 1471, 1609, 1889, 2029, 2099, 2239, 2309, 2311, 2381, 2521, 2591, 2659, 2729, 2731, 2801, 2939, 3011, 3079, 3221, 3359, 3361

Offset: 1

Giorgio Balzarotti and Paolo P. Lava, Oct 13 2005

For every prime p here, the cyclotomic polynomial Phi(35p,x) is flat.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A117223.

[p: p in PrimesUpTo(4000) | p mod 70 in [1, 69]]; // Vincenzo Librandi, Aug 14 2012

Select[Prime[Range[500]], MemberQ[{1, 69}, Mod[#, 70]] &] (* Vincenzo Librandi, Aug 14 2012 *)

Edited by T. D. Noe, May 19 2009

A136418 Smallest order of the cyclotomic polynomial whose maximal coefficient in absolute value is n.

Original entry on oeis.org

0, 105, 385, 1365, 1785, 2805, 3135, 10353, 6545, 12155, 21385, 11165, 21505, 10465, 16555, 19285, 37961, 35105, 18445, 24395, 23205, 53669, 11305, 28595, 17255, 36465, 20615, 42315, 123585, 31535, 49335, 39585, 61295, 35805, 72709, 54285

Offset: 1

Robert G. Wilson v, Mar 31 2008

nonn

This differs from A013594.
For squarefree k, are there an infinite number of cyclotomic polynomials Phi(k,x) of height n? This is true for n=1 because it is known that there are an infinite number of flat cyclotomic polynomials with k the product of three distinct primes. See A117223. - T. D. Noe, Apr 22 2008
There are an infinite number of cyclotomic polynomials of height n if the following generalization of Kaplan's theorem 2 is true: Let N be the product of distinct odd primes and let p be one of those primes. Let q any prime such that q = p (mod N/p), then the height of Phi(Nq/p,x) is the same as the height of Phi(N,x). By Dirichlet's theorem, there are an infinite number of primes q. [From T. D. Noe, Apr 13 2010]

T. D. Noe, Table of n, a(n) for n=1..1000
Nathan Kaplan, Flat cyclotomic polynomials of order three, J. Number Theory 127 (2007), 118-126.

Cf. A013594, A046887, A134518.

f[n_] := f[n] = Max@ Abs@ CoefficientList[ Cyclotomic[n, x], x]; Do[ f@n, {n, 100000}]; t = Array[f, 31000]; Table[ Position[t, n, 1, 1], {n, 25}]//Flatten

More terms from T. D. Noe, Apr 22 2008

A161504 Primes congruent to {1, 2, 10, 11, 19, 20} mod 21.

Original entry on oeis.org

2, 11, 19, 23, 31, 41, 43, 53, 61, 73, 83, 103, 107, 127, 137, 149, 157, 167, 179, 191, 199, 211, 229, 233, 241, 251, 263, 271, 283, 293, 313, 317, 337, 347, 359, 367, 379, 389, 397, 401, 409, 419, 421, 431, 439, 443, 461, 463, 503, 523, 547, 557, 569, 577, 587

Offset: 1

T. D. Noe, Jun 17 2009

nonn

The cyclotomic polynomial Phi(21p,x) is flat only for p in this sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1001 [a(1)=2 inserted by _Georg Fischer_, Jul 26 2020]

Cf. A117223, A124826, A141888, A141892, A141893, A141897, A141898, A160495.

Flatten[Table[Select[21*n+{-10,-2,-1,1,2,10}, PrimeQ], {n,50}]]
Select[Prime[Range[2,250]],MemberQ[{1,2,10,11,19,20},Mod[#,21]]&] (* Harvey P. Dale, May 03 2019 *)

Union of A124826, A141888, A141892, A141893, A141897, A141898.

a(1)=2 inserted by Georg Fischer, Jul 26 2020

A161505 Primes congruent to {1, 7, 8, 25, 26, 32} mod 33.

Original entry on oeis.org

7, 41, 59, 67, 73, 107, 131, 139, 157, 173, 191, 197, 199, 223, 239, 257, 263, 271, 331, 337, 389, 397, 421, 461, 463, 487, 503, 521, 569, 587, 593, 601, 619, 653, 659, 661, 701, 719, 727, 733, 751, 857, 859, 883, 983, 991, 997, 1031, 1049, 1063, 1097, 1123

Offset: 1

T. D. Noe, Jun 17 2009

nonn

The cyclotomic polynomial Phi(33p,x) is flat only for p in this sequence.

Georg Fischer, Table of n, a(n) for n = 1..1000

Cf. A117223, A142049, A142053, A142054, A142063, A142064, A142068, A160495.

Flatten[Table[Select[33*n+{-8,-7,-1,1,7,8}, PrimeQ], {n,50}]]

Union of A142049, A142053, A142054, A142063, A142064, A142068.

a(1)=7 inserted by Georg Fischer, Jul 26 2020

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).

A160493 Maximum height of the third-order cyclotomic polynomial Phi(pqr,x) with p

Original entry on oeis.org

2, 2, 2, 3, 2, 2, 3, 2, 3, 2, 4, 3, 2, 3, 2, 3, 2, 3, 4, 2, 2, 4, 2, 6, 3, 3, 2, 4, 2, 2, 3, 5, 2, 4, 3, 7, 2, 3, 4, 2, 7, 3, 2, 5, 2, 3, 4, 3, 2, 4, 2, 3, 7, 4, 2, 3, 2, 7, 2, 9, 2, 4, 3, 2, 6, 3, 3, 4, 7, 2, 7, 2, 3, 8, 6, 2, 4, 3, 2, 4, 11, 3, 2, 7, 2, 4, 2, 5, 7, 3, 2, 10, 4, 2, 3, 4, 3, 6, 2, 9

Offset: 1

T. D. Noe, May 15 2009

nonn

The height of a polynomial is the maximum of the absolute value of its coefficients. Sequence A046388 gives increasing values of pq. As proved by Kaplan, to compute the maximum height of Phi(pqr,x) for any prime r, there are only (p-1)(q-1)/2 values of r to consider. The set s of values of r can be taken to be primes greater than q such that the union of s and -s (mod pq) contains every number less than and coprime to pq. It appears that when p=3, the maximum height is 2; when p=5, the maximum is 3; when p=7, the maximum is 3 or 4; and when p=11, the maximum is no greater than 7.

Nathan Kaplan, Flat cyclotomic polynomials of order three, J. Number Theory 127 (2007), 118-126.

Cf. A046388, A117223.

a(n) = maximum height of Phi(A046388(n)*r,x) for any prime r>q.

A160494 Least prime r > q such that the third-order cyclotomic polynomial Phi(pqr,x) is flat with p,q,r distinct odd primes, ordered by pq.

Original entry on oeis.org

29, 11, 41, 71, 17, 23, 53, 23, 131, 41, 307, 509, 61, 181, 37, 191, 41, 229, 239, 89, 47, 797, 73, 571, 499, 157, 59, 643, 73, 71, 739, 373, 71, 607, 359, 419, 83, 431, 433, 89, 443, 941, 83, 1481, 109, 251, 1553, 1061, 101, 1721, 101, 401, 599, 251, 131

Offset: 1

T. D. Noe, May 15 2009

nonn

A flat polynomial is defined to be a polynomial whose coefficients are -1, 0, or 1. Sequence A046388 gives the product pq. As proved by Kaplan, given odd primes p < q, it is always possible to find a prime r > q such that Phi(pqr,x) is flat.

			a(1)=29 because 15*29 is the least multiple of 15 that produces a flat cyclotomic polynomial.

Nathan Kaplan, Flat cyclotomic polynomials of order three, J. Number Theory 127 (2007), 118-126.

Cf. A046388, A117223.

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A160355 Odd indices pqr of flat cyclotomic polynomials of order 3 which are not of the form r = +/-1 (mod pq).

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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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A111092 Primes congruent to {1,69} mod 70.

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A136418 Smallest order of the cyclotomic polynomial whose maximal coefficient in absolute value is n.

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A161504 Primes congruent to {1, 2, 10, 11, 19, 20} mod 21.

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A161505 Primes congruent to {1, 7, 8, 25, 26, 32} mod 33.

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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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A160493 Maximum height of the third-order cyclotomic polynomial Phi(pqr,x) with p

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A160494 Least prime r > q such that the third-order cyclotomic polynomial Phi(pqr,x) is flat with p,q,r distinct odd primes, ordered by pq.

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