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A072226
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Numbers n such that the n-th cyclotomic polynomial evaluated at 2 (=A019320(n)) is prime.
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19
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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, 38, 40, 42, 46, 49, 56, 61, 62, 65, 69, 77, 78, 80, 85, 86, 89, 90, 93, 98, 107, 120, 122, 126, 127, 129, 133, 145, 150, 158, 165, 170, 174, 184, 192, 195, 202, 208, 234, 254, 261
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OFFSET
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1,1
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COMMENTS
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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
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
Conjecture: no term greater than 12 can be congruent to 4 modulo 8. 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
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REFERENCES
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Yves Gallot, Cyclotomic polynomials and prime numbers (November 12, 2000; revised January 5, 2001)
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LINKS
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MATHEMATICA
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Select[Range[600], PrimeQ[Cyclotomic[ #, 2]]&]
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PROG
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(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
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CROSSREFS
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Corresponding primes are listed in A292015.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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