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

%I A138934 #42 Sep 10 2024 16:10:25
%S A138934 1,2,4,6,8,12,16,20,28,40,60,92,96,104,140,148,156,300,356,408,596,
%T A138934 612,692,732,756,800,952,996,1004,1228,1268,2240,2532,3060,3796,3824,
%U A138934 3944,5096,5540,7476,7700,8544,9800,14628,15828,16908,18480,20260,21924,24656,38456
%N A138934 Indices k such that A019322(k) = Phi[k](4) is prime, where Phi is a cyclotomic polynomial.
%C A138934 It appears that except for 1,2 and 6, all terms of this sequence are multiples of 4.
%C A138934 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
%C A138934 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
%H A138934 Robert Price, <a href="/A138934/b138934.txt">Table of n, a(n) for n = 1..51</a>
%H A138934 Yves Gallot, <a href="http://yves.gallot.pagesperso-orange.fr/papers/cyclotomic.html">Cyclotomic polynomials and prime numbers</a>
%H A138934 <a href="/index/Cy#CyclotomicPolynomialsValuesAtX">Index entries for cyclotomic polynomials, values at X</a>
%t A138934 Select[Range[1000], PrimeQ[Cyclotomic[#, 4]] &]
%o A138934 (PARI) for( i=1,999, ispseudoprime( polcyclo(i,4)) && print1( i","))
%Y A138934 Cf. A019322, A072226, A138920-A138940.
%K A138934 nonn
%O A138934 1,2
%A A138934 _M. F. Hasler_, Apr 03 2008
%E A138934 a(29)-a(51) from _Robert Price_, Apr 12 2012