A160350 -id:A160350 - 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

A160353 Numbers of the form pqr, 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

M. F. Hasler, May 11 2009

nonn

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.

			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.

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

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","))

A160354 Indices pqr of flat cyclotomic polynomials of order 3 which are not of the form r = +/-1 (mod pq).

Original entry on oeis.org

70, 130, 154, 170, 230, 231, 238, 266, 286, 322, 370, 374, 399, 418, 430, 434, 442, 470, 483, 494, 518, 530, 598, 638, 646, 651, 658, 663, 670, 682, 730, 741, 742, 754, 782, 806, 814, 826, 830, 854, 874, 902, 938, 962, 970, 986, 1022, 1030, 1034, 1054, 1066

Offset: 1

M. F. Hasler, May 11 2009

nonn

Kaplan (2007) has shown that Phi(pqr) has coefficients in {0,1,-1} if r = +-1 (mod pq), where pA160350 which do not satisfy this equality.

Yet most elements are even, i.e. in A075819. Sequence A160355 is the subsequence of odd terms. See A160350 for more details.

			a(1)=70=2*5*7 is the smallest element of A160350 for which the largest factor (7) is not congruent to +- 1 modulo the product of the smaller factors (2*5).

Robin Visser, Table of n, a(n) for n = 1..10000
Nathan Kaplan, Flat cyclotomic polynomials of order three, J. Number Theory 127 (2007), 118-126.

for( pqr=1,1999, 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 A160350 \ A160352.

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

Original entry on oeis.org

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.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Programs

PARI

A160354 Indices pqr of flat cyclotomic polynomials of order 3 which are not of the form r = +/-1 (mod pq).

Views

Author

Keywords

Comments

Examples

Links

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

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