A138934 Indices k such that A019322(k) = Phi[k](4) is prime, where Phi is a cyclotomic polynomial.

1, 2, 4, 6, 8, 12, 16, 20, 28, 40, 60, 92, 96, 104, 140, 148, 156, 300, 356, 408, 596, 612, 692, 732, 756, 800, 952, 996, 1004, 1228, 1268, 2240, 2532, 3060, 3796, 3824, 3944, 5096, 5540, 7476, 7700, 8544, 9800, 14628, 15828, 16908, 18480, 20260, 21924, 24656, 38456

Offset: 1

Author

M. F. Hasler, Apr 03 2008

Keywords

• nonn

Comments

It appears that except for 1,2 and 6, all terms of this sequence are multiples of 4.

It also appears that all cyclotomic polynomials, Phi(k,x), where k is a multiple of 4 have no odd powers of x. For example, Phi(20,x) = x^8 - x^6 + x^4 - x^2 + 1. This implies that Phi(k,x) = Phi(k,-x), where k is a multiple of 4. - Robert Price, Apr 13 2012

Second comment is true; this follows from applying Theorem 1.1 in the Gallot paper with p = 2 and m even. - Charlie Neder, May 16 2019

Links

• Robert Price, Table of n, a(n) for n = 1..51
• Yves Gallot, Cyclotomic polynomials and prime numbers
• Index entries for cyclotomic polynomials, values at X

Crossrefs

Cf. A019322, A072226, A138920-A138940.

Programs

Mathematica

Select[Range[1000], PrimeQ[Cyclotomic[#, 4]] &]

PARI

for( i=1,999, ispseudoprime( polcyclo(i,4)) && print1( i","))

Extensions

a(29)-a(51) from Robert Price, Apr 12 2012