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.

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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.
		

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","))

A159908 Number of pairs (p,q) of 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

1, 3, 6, 9, 13, 15, 19, 23, 27, 30, 34, 35, 43, 40, 45, 47, 54, 57, 58, 64, 69, 69, 71, 79, 79, 84, 86, 87, 97, 96, 94, 107, 106, 109, 120, 111, 120, 123, 124, 133, 135, 134, 144, 143, 143, 154, 154, 154, 163, 161, 167, 175, 174, 175, 179, 183, 187, 191, 193, 199, 197, 202, 203
Offset: 1

Views

Author

M. F. Hasler, May 09 2009

Keywords

Comments

The cyclotomic polynomial Phi[pqr] (p,q,r primes) can only have coefficients with absolute value > 1 if p,q,r are distinct odd primes. This sequence also counts the trivial cases where (1): p=2, or (2): p=q, or (3): q=r. The number of these cases is A008486(n-1). Sequence A159909 counts only the nontrivial cases.

Crossrefs

Programs

  • PARI
    A159908(n) = sum( i=1,n, my(pq=prime(n)*prime(i)); sum( j=1,i, vecmax(abs(Vec(polcyclo(prime(j)*pq))))==1 ))

Formula

a(n) = A008486(n-1) + A159909(n).

Extensions

More terms from Robin Visser, Aug 09 2023
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