cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 19 results. Next

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

Views

Author

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

Keywords

Comments

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)

Links

Crossrefs

Cf. A013594, A117223, A117318, A137979.
Flat cyclotomic polynomial: A117223 (order 3), A117318 (order 4).

Programs

  • 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;
end do: # R. J. Mathar, Nov 28 2016
  • Mathematica
    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 *)
  • 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

Extensions

Definition clarified by Harvey P. Dale, Mar 13 2013
New name from Michel Marcus, Apr 29 2018

A117318 Numbers n such that Phi(n,x) is a flat cyclotomic polynomial of order four.

Original entry on oeis.org

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
Offset: 1

Views

Author

T. D. Noe, Mar 07 2006

Keywords

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
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]

Links

Crossrefs

Cf. A117223 (third-order flat cyclotomic polynomials).

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

Views

Author

T. D. Noe, Dec 16 2008

Keywords

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.

Links

Crossrefs

Cf. A152940, A152941, A152942, A152943.

Programs

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

Views

Author

T. D. Noe, Jan 09 2009

Keywords

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.

Links

Crossrefs

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

Programs

  • Mathematica
    Select[Range[2000], OddQ[#] && SquareFreeQ[#] && Max[ Abs[ CoefficientList[ Cyclotomic[#, x], x]]] > 1&] (* Jean-François Alcover, Nov 14 2016 *)
  • PARI
    is(n)=issquarefree(n) && n%2 && vecmax(abs(Vec(polcyclo(n))))>1 \\ Charles R Greathouse IV, Nov 05 2017

A160350 Indices n=pqr of flat cyclotomic polynomials, where p

Original entry on oeis.org

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
Offset: 1

Views

Author

M. F. Hasler, May 11 2009, May 14 2009

Keywords

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.

Examples

			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.

Links

Crossrefs

Cf. A159908, A159909 (counts (p, q) for given r).

Programs

  • PARI
    for( pqr=1,999, my(f=factor(pqr)); #f~==3 & vecmax(f[,2])==1 & vecmax(abs(Vec(polcyclo(pqr))))==1 & print1(pqr","))

A160353 Numbers of the form p*q*r, where p < q < r are odd primes such that r = +/-1 (mod p*q).

Original entry on oeis.org

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
Offset: 1

Views

Author

M. F. Hasler, May 11 2009

Keywords

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.

Examples

			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.

Links

Programs

  • 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","))

A160495 Irregular triangle of residue classes (mod pq) of primes r such that the cyclotomic polynomial Phi(pqr,x) is flat.

Original entry on oeis.org

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
Offset: 1

Views

Author

T. D. Noe, May 15 2009

Keywords

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?

Examples

			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.

Links

Crossrefs

Cf. A046388, A117223.

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

Views

Author

Max Alekseyev, May 19 2006

Keywords

Comments

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

Links

Crossrefs

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

A132240 Primes congruent to {1, 29} mod 30.

Original entry on oeis.org

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
Offset: 1

Views

Author

Omar E. Pol, Aug 15 2007

Keywords

Comments

For every prime p here, the cyclotomic polynomial Phi(15p,x) is flat.
Primes in A175887. [Reinhard Zumkeller, Jan 07 2012]

Links

Crossrefs

Cf. A000040, A001097, A001359, A006512, A039949, A057204, A068228, A087715, A129805, A117223, A010051.

Programs

  • Haskell
    a132240 n = a132240_list !! (n-1)
a132240_list = [x | x <- a175887_list, a010051 x == 1]
-- Reinhard Zumkeller, Jan 07 2012
  • Magma
    [ p: p in PrimesUpTo(1300) | p mod 30 in {1, 29} ]; // Vincenzo Librandi, Aug 14 2012
  • 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 *)

A159909 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.

Original entry on oeis.org

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
Offset: 1

Views

Author

M. F. Hasler, May 09 2009

Keywords

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.

Examples

			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).

Links

Crossrefs

Cf. A117223. [T. D. Noe, May 11 2009]

Programs

  • 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 ))

Extensions

Extended by T. D. Noe, May 11 2009
More terms from Robin Visser, Aug 09 2023
Showing 1-10 of 19 results. Next

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