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A072226 Numbers k such that the k-th cyclotomic polynomial evaluated at 2 (=A019320(k)) is prime.

%I A072226 #46 Dec 03 2024 20:03:46
%S A072226 2,3,4,5,6,7,8,9,10,12,13,14,15,16,17,19,22,24,26,27,30,31,32,33,34,
%T A072226 38,40,42,46,49,56,61,62,65,69,77,78,80,85,86,89,90,93,98,107,120,122,
%U A072226 126,127,129,133,145,150,158,165,170,174,184,192,195,202,208,234,254,261
%N A072226 Numbers k such that the k-th cyclotomic polynomial evaluated at 2 (=A019320(k)) is prime.
%C A072226 The prime n in this sequence are in A000043, which produce the Mersenne primes. If 2p is in this sequence, with p prime, then p is a Wagstaff number, A000978. - _T. D. Noe_, Apr 02 2008
%C A072226 While the sequence looks quite dense for small values, note that there are only 10 values in the interval [700,1200]. - _M. F. Hasler_, Apr 03 2008
%C A072226 No term greater than 12 can be congruent to 4 modulo 8 as proved by Schinzel (1962), see also Pomerance (2024). Note the Aurifeuillean factorization: Product_{4|d, d|8*k+4} Phi(d,2) = 2^(4k+2) + 1 = (2^(2k+1) - 2^(k+1) + 1) * (2^(2k+1) + 2^(k+1) + 1). If Phi(8*k+4,2) is prime, then it divides either 2^(2k+1) - 2^(k+1) + 1 or 2^(2k+1) + 2^(k+1) + 1. This immediately proves that no term can be of the form 4*p for odd primes p >= 5 Since Phi(4*p,2) = (2^(2*p) + 1)/5. - _Jianing Song_, Apr 04 2022; edited by _Max Alekseyev_, Dec 03 2024
%H A072226 Max Alekseyev, <a href="/A072226/b072226.txt">Table of n, a(n) for n = 1..289</a> (terms 1..234, 235..277, and 278..289 from Yves Gallot, T. D. Noe, and Carl Pomerance, respectively)
%H A072226 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>
%H A072226 Yves Gallot, <a href="https://web.archive.org/web/20231001175553/https://yves.gallot.pagesperso-orange.fr/papers/cyclotomic.pdf">Cyclotomic polynomials and prime numbers</a>
%H A072226 Carl Pomerance, <a href="https://arxiv.org/abs/2411.04213">Cyclotomic primes</a>, arXiv:2411.04213 [math.NT], 2024.
%H A072226 Wikipedia, <a href="https://en.wikipedia.org/wiki/Aurifeuillean_factorization">Aurifeuillean factorization</a>
%H A072226 <a href="/index/Cy#CyclotomicPolynomialsValuesAtX">Index entries for cyclotomic polynomials, values at X</a>
%t A072226 Select[Range[600], PrimeQ[Cyclotomic[ #, 2]]&]
%o A072226 (PARI) for( i=1,999, ispseudoprime( polcyclo(i,2)) && print1( i",")) /* for PARI < 2.4.2 use ...subst(polcyclo(i),x,2)... */ \\ _M. F. Hasler_, Apr 03 2008
%Y A072226 Corresponding primes are listed in A292015.
%Y A072226 Cf. A138920-A138940.
%K A072226 nonn
%O A072226 1,1
%A A072226 _Reiner Martin_, Jul 04 2002
%E A072226 Edited by _Max Alekseyev_, Apr 25 2018