Search: a070522 -id:a070522
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A070523
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Numbers k such that cyclotomic(k, prime(k)) is a prime number.
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+10
3
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3, 6, 7, 14, 19, 31, 34, 66, 93, 307, 402, 421, 600, 848, 1022, 1057, 1906, 3772, 4184, 4364
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OFFSET
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1,1
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COMMENTS
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Values corresponding to a(3)=7 through a(13)=600 have been certified prime with Primo. Their sizes in decimal digits are 8, 10, 33, 64, 35, 51, 162, 1012, 455, 1455 and 584, respectively. Values corresponding to a(14) through a(17) are probable primes with lengths 1588, 1689, 3535 and 4014 decimal digits. a(18)>2000. - Rick L. Shepherd, Jul 10 2002
All terms <= 1906 have been proven with PARI's ECPP. No other terms <= 20000. - Lucas A. Brown, Jan 03 2021
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LINKS
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EXAMPLE
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For n=7: 1+x+x^2+x^3+x^4+x^5+x^6 at x=prime(7)=17 gives a prime 25646167.
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PROG
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(PARI) for(n=1, 2000, if(isprime(eval(polcyclo(n, prime(n)))), print1(n, ", ")))
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A070525
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Numbers n such that n-th cyclotomic polynomial evaluated at phi(n) is a prime number.
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+10
3
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2, 3, 4, 6, 7, 8, 12, 18, 21, 30, 45, 48, 70, 120, 127, 153, 182, 204, 212, 282, 318, 322, 910, 1167, 1177, 1342, 1680, 1963, 2670, 4398, 4655, 8088, 8599, 8808, 19680
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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These are probable primes for n > 910. No others for n <= 10000. The prime values of n are 2, 3, 7, 127 and 8599 (A088856). - T. D. Noe, Nov 23 2003
All terms <= 2670, except 1963, have been certified prime with PARI's ECPP. There are no other terms <= 25000. - Lucas A. Brown, Jan 08 2021
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LINKS
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EXAMPLE
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n=7: Phi(7)=6, Cyclotomic(7,6)=1+6+36+216+1296+7776+46656=55987 is prime.
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MATHEMATICA
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Do[s=Cyclotomic[n, EulerPhi[n]]; If[PrimeQ[s], Print[n]], {n, 1, 400}]
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PROG
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(PARI) isok(n) = isprime(polcyclo(n, eulerphi(n))); \\ Michel Marcus, Sep 01 2019
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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