Displaying 1-10 of 19 results found.
Numbers k such that Phi(k,x) is a cyclotomic polynomial containing a coefficient with an absolute value greater than one.
+10
11
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
COMMENTS
Previous name was: Orders of cyclotomic polynomials containing a coefficient with an absolute value greater than one.
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)
MAPLE
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;
MATHEMATICA
S[ n_ ] := For[ j=1; t=0, j<n, j++, t=Cases[ CoefficientList[ Cyclotomic[ j, x ], x ], k_ /; Abs[ k ]>1 ]; 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 *)
PROG
(PARI) 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
AUTHOR
Peter T. Wang (peterw(AT)cco.caltech.edu)
Numbers n such that Phi(n,x) is a flat cyclotomic polynomial of order four.
+10
8
431985, 757335, 865365, 1134915, 1296885, 1297815, 1675365, 1729335, 1891815, 2161785, 2162715, 2595165, 2648715, 2649585, 3027165, 3028035, 3132015, 3133785, 3347985, 3405615, 3565785, 3784065, 3891585, 4698465, 4920285, 5188935, 5189865, 5676315
COMMENTS
A flat polynomial is defined to be a polynomial whose coefficients are -1, 0, or 1. Order four means that n is the product of four odd primes p<q<r<s.
For pqrs to be flat, it appears that three conditions on p < q < r < s are required: q = -1 (mod p), r = +-1 (mod pq), and s = +-1 (mod pqr). [ T. D. Noe, Apr 13 2010]
CROSSREFS
Cf. A117223 (third-order flat cyclotomic polynomials).
Odd squarefree numbers n such that the cyclotomic polynomial Phi(n,x) has height 1.
+10
8
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
COMMENTS
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.
Odd squarefree numbers n such that the cyclotomic polynomial Phi(n,x) has height > 1.
+10
8
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
COMMENTS
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.
MATHEMATICA
Select[Range[2000], OddQ[#] && SquareFreeQ[#] && Max[ Abs[ CoefficientList[ Cyclotomic[#, x], x]]] > 1&] (* Jean-François Alcover, Nov 14 2016 *)
Indices n=pqr of flat cyclotomic polynomials, where p<q<r are primes.
+10
5
30, 42, 66, 70, 78, 102, 110, 114, 130, 138, 154, 170, 174, 182, 186, 190, 222, 230, 231, 238, 246, 258, 266, 282, 286, 290, 310, 318, 322, 354, 366, 370, 374, 399, 402, 406, 410, 418, 426, 430, 434, 435, 438, 442, 465, 470, 474, 483, 494, 498, 506, 518, 530
COMMENTS
A polynomial is called flat iff it is of height 1, where the height is the largest absolute value of the coefficients.
A cyclotomic polynomial phi(n) is said of order 3 iff n=pqr with distinct (usually odd) primes p,q,r.
It is well known that phi(n) is flat if n has less than 3 odd prime factors, so this sequence includes all numbers of the form 2pq, with primes q>p>2, i.e. A075819. Sequence A117223 lists the complement, i.e. odd terms in this sequence, which start with 231 = 3*7*11. Moreover, Kaplan shows that the present sequence also includes pqr if r = +-1 (mod pq). Sequence A160352 lists the subsequence of all such numbers, while A160354 lists elements which are not of this form.
EXAMPLE
a(1)=30=2*3*5 is the smallest product of three distinct primes, and Phi[30] = X^8 + X^7 - X^5 - X^4 - X^3 + X + 1 has only coefficients in {0,1,-1}.
a(19)=231=3*7*11 is the smallest odd product of three distinct primes p,q,r such that Phi[pqr] is flat.
PROG
(PARI) for( pqr=1, 999, my(f=factor(pqr)); #f~==3 & vecmax(f[, 2])==1 & vecmax(abs(Vec(polcyclo(pqr))))==1 & print1(pqr", "))
Numbers of the form p*q*r, where p < q < r are odd primes such that r = +/-1 (mod p*q).
+10
4
435, 465, 861, 885, 903, 915, 1335, 1743, 2211, 2235, 2265, 2485, 2667, 2685, 2715, 3081, 3165, 3507, 3585, 3615, 4035, 4065, 4323, 4431, 4865, 4965, 5151, 5253, 5271, 5385, 5835, 5995, 6123, 6153, 6285, 6315, 6441, 6501, 6567, 6735, 7077, 7185, 7385
COMMENTS
Kaplan (2007) has shown that this is a subsequence of A117223 (and thus of A160350; see there for the reference), i.e., the cyclotomic polynomial phi(n) has coefficients in {0,1,-1} for indices n listed here. This is a subsequence of A160352 which drops the requirement that p > 2. See A160350 for further details and references.
EXAMPLE
a(1) = 435 = 3*5*29 is the smallest product of odd primes p < q < r such that r is congruent to +/- 1 modulo the product of the smaller factors, p*q.
PROG
(PARI) forstep( pqr=1, 9999, 2, my(f=factor(pqr)); #f~==3 & vecmax(f[, 2])==1 & abs((f[3, 1]+1)%(f[1, 1]*f[2, 1])-1)==1 & print1(pqr", "))
Irregular triangle of residue classes (mod pq) of primes r such that the cyclotomic polynomial Phi(pqr,x) is flat.
+10
4
1, 14, 1, 2, 10, 11, 19, 20, 1, 7, 8, 25, 26, 32, 1, 34, 1, 2, 8, 17, 19, 20, 22, 31, 37, 38, 1, 13, 20, 22, 23, 28, 29, 31, 38, 50, 1, 2, 53, 54, 1, 2, 7, 13, 16, 23, 28, 29, 34, 41, 44, 50, 55, 56, 1, 64, 1, 7, 8, 10, 17, 19, 28, 41, 50, 52, 59, 61, 62, 68
COMMENTS
A polynomial is flat if its coefficients are 1, 0, or -1. The values of pq are in sequence A046388. Each row begins with 1 and ends with pq-1. For each number k in a row, the number pq-k is also in the row. Row n has 2* A160496(n) terms. For the pq in sequence A160497, the row has only two terms. By Kaplan's theorems 2 and 3, only the first prime r in each residue class 1..(p-1)(q-1)/2 needs to be checked to determine whether the residue class produces flat cyclotomic polynomials. Is there a simplier method of finding these residue classes?
EXAMPLE
The second row (1,2,10,11,19,20) is for pq=21. If r is a prime with r mod pq equal to one of these 6 values, then Phi(21*r,x) is flat.
Primitive orders of cyclotomic polynomials containing a coefficient with absolute value >= 2.
+10
3
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
Primes congruent to {1, 29} mod 30.
+10
3
29, 31, 59, 61, 89, 149, 151, 179, 181, 211, 239, 241, 269, 271, 331, 359, 389, 419, 421, 449, 479, 509, 541, 569, 571, 599, 601, 631, 659, 661, 691, 719, 751, 809, 811, 839, 929, 991, 1019, 1021, 1049, 1051, 1109, 1171, 1201, 1229, 1231
COMMENTS
For every prime p here, the cyclotomic polynomial Phi(15p,x) is flat.
MATHEMATICA
Select[Prime[Range[1000]], MemberQ[{1, 29}, Mod[#, 30]]&] (* Vincenzo Librandi, Aug 14 2012 *) Select[Flatten[#+{1, 29}&/@(30Range[0, 50])], PrimeQ] (* Harvey P. Dale, Sep 08 2021 *)
PROG
(Haskell)
a132240 n = a132240_list !! (n-1)
a132240_list = [x | x <- a175887_list, a010051 x == 1]
(Magma) [ p: p in PrimesUpTo(1300) | p mod 30 in {1, 29} ]; // Vincenzo Librandi, Aug 14 2012
CROSSREFS
Cf. A000040, A001097, A001359, A006512, A039949, A057204, A068228, A087715, A129805, A117223, A010051.
Number of pairs (p,q) of odd primes p < q < r=prime(n) such that the cyclotomic polynomial Phi(p*q*r) has no coefficient > 1 in absolute value.
+10
3
0, 0, 0, 0, 1, 0, 1, 2, 3, 3, 4, 2, 7, 1, 3, 2, 6, 6, 4, 7, 9, 6, 5, 10, 7, 9, 8, 6, 13, 9, 4, 14, 10, 10, 18, 6, 12, 12, 10, 16, 15, 11, 18, 14, 11, 19, 16, 13, 19, 14, 17, 22, 18, 16, 17, 18, 19, 20, 19, 22, 17, 19, 17, 19, 19, 19, 31, 25, 13, 38, 20, 23, 25, 23, 31, 30, 31, 19
COMMENTS
The cyclotomic polynomial Phi[pqr] can only have coefficients with absolute value > 1 if p,q,r are distinct odd primes, that's why we require 2 < p < q < r. If any of these inequalities is replaced by equality, then Phi[pqr] necessarily has only zero or unit (+-1) coefficients. Sequence A159908 counts all possibilities including these trivial cases.
LINKS
Phil Carmody, David Broadhurst, Maximilian Hasler, Makoto Kamada, Cyclotomic polynomial puzzles, digest of 43 messages in primenumbers Yahoo group, May 9, 2009 - May 23, 2013.
EXAMPLE
a(5)=1 is the first nonzero term, since the smallest example for Phi(pqr) having no coefficient > 1 (in abs. value) for odd primes p<q<r is obtained for r=prime(5), namely Phi(3*7*11).
PROG
(PARI) A159909(n) = sum( i=2, n-1, my(pq=prime(n)*prime(i)); sum( j=2, i-1, vecmax(abs(Vec(polcyclo(prime(j)*pq))))==1 ))
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