A250177 -id:A250177 - cp's OEIS frontend

A100330 Positive integers k such that k^6 + k^5 + k^4 + k^3 + k^2 + k + 1 is prime.

1, 2, 3, 5, 6, 13, 14, 17, 26, 31, 38, 40, 46, 56, 60, 61, 66, 68, 72, 73, 80, 87, 89, 93, 95, 115, 122, 126, 128, 146, 149, 156, 158, 160, 163, 180, 186, 192, 203, 206, 208, 220, 221, 235, 237, 238, 251, 264, 266, 280, 282, 290, 294, 300, 303, 320, 341, 349, 350

Offset: 1

Ray G. Opao, Nov 16 2004

nonn

The corresponding primes are A088550. - Bernard Schott, Dec 20 2012

k = 5978493 * 2^150006 - 1 is an example of a very large term of this sequence. The generated prime is proved by the N-1 method (because k is prime and k*(k+1) is fully factored and this provides for an exactly 33.33...% factorization for Phi_7(k) - 1). - Serge Batalov, Mar 13 2015

			2 is in the sequence because 2^6 + 2^5 + 2^4 + 2^3 + 2^2 + 2 + 1 = 127, which is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A100331, A250174 (Phi_14(n) = n^6 - n^5 + n^4 - n^3 + n^2 - n + 1 primes; these two sequences can also be considered an extension of each other into negative n values), A250177 (Phi_21(n) primes).

[n: n in [1..500]| IsPrime(n^6 + n^5 + n^4 + n^3 + n^2 + n + 1)]; // Vincenzo Librandi, Feb 08 2014

A100330 := proc(n)
    option remember;
    local a;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isprime(numtheory[cyclotomic](7,a)) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A100330(n),n=1..30) ; # R. J. Mathar, Feb 07 2014

Select[Range[350], PrimeQ[Sum[ #^i, {i, 0, 6}]] &] (* Ray Chandler, Nov 17 2004 *)
Do[If[PrimeQ[n^6 + n^5 + n^4 + n^3 + n^2 + n + 1], Print[n]], {n, 1, 500}] (* Vincenzo Librandi, Feb 08 2014 *)

is(n)=isprime(polcyclo(7,n)) \\ Charles R Greathouse IV, Apr 28 2015

A250174 Numbers n such that Phi_14(n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

2, 3, 10, 11, 14, 15, 16, 17, 18, 21, 24, 25, 29, 37, 43, 44, 46, 49, 52, 54, 61, 66, 72, 73, 78, 84, 86, 87, 99, 101, 106, 114, 115, 128, 133, 135, 136, 143, 145, 148, 164, 169, 170, 173, 200, 219, 224, 226, 228, 231, 234, 240, 248, 255, 262, 275, 281, 282, 298, 301

Offset: 1

Eric Chen, Dec 24 2014

nonn

n = 9069 * 2^64163 + 1 is an example of a rather large member of this sequence. The generated 115914 decimal digit prime is proved by the N-1 method (because n is prime and n*(n-1) is fully factored and this provides for an exactly 33.33...% factorization for Phi_14(n) - 1). - Serge Batalov, Mar 13 2015

			2 is in the sequence because 2^6-2^5+2^4-2^3+2^2-2+1 = 43 which is prime.

Serge Batalov, Table of n, a(n) for n = 1..1595

See A250177 for cross-references, A100330 (Phi_7(n) = n^6 + n^5 + n^4 + n^3 + n^2 + n + 1 primes; these two sequences can also be considered an extension of each other into negative n values), A250177 (Phi_21(n) primes).

a250174[n_] := Select[Range[n], PrimeQ@Cyclotomic[14, #] &]; a250174[256]

isok(n) = isprime(polcyclo(14, n)); \\ Michel Marcus, Mar 13 2015

A250178 Numbers k such that Phi_22(k) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

2, 6, 12, 13, 23, 24, 26, 35, 62, 69, 91, 105, 147, 160, 163, 183, 185, 193, 229, 232, 233, 236, 248, 262, 269, 280, 294, 303, 315, 330, 376, 394, 426, 430, 440, 464, 469, 491, 492, 508, 537, 625, 629, 647, 653, 666, 752, 772, 775, 786, 788, 832, 840, 852, 854, 855, 859, 905, 922, 972, 993, 998

Offset: 1

Eric Chen, Dec 24 2014

nonn

Numbers k such that (k^11+1)/(k+1) is a prime. - Vincenzo Librandi, May 21 2018

G. C. Greubel, Table of n, a(n) for n = 1..10000

See A250177 for references.

[n: n in [1..1000]| IsPrime((n^11+1) div (n+1))]; // Vincenzo Librandi, May 21 2018

a250178[n_] := Select[Range[n], PrimeQ@Cyclotomic[22, #] &]; a250178[1000] (* Michael De Vlieger, Dec 25 2014 *)

isok(n) = isprime(polcyclo(22, n)); \\ Michel Marcus, Sep 29 2015

A250179 Numbers n such that Phi_24(n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

2, 3, 5, 6, 10, 13, 14, 15, 18, 25, 26, 27, 31, 34, 37, 39, 40, 42, 44, 46, 49, 50, 53, 59, 62, 63, 65, 68, 69, 75, 76, 77, 83, 87, 99, 100, 102, 104, 107, 110, 142, 152, 161, 166, 174, 175, 184, 187, 188, 191, 192, 199, 204, 207, 208, 213, 215, 222, 224, 227, 238, 244, 245, 247, 250, 256, 259, 263, 264, 265, 268, 274, 279, 286, 289, 303, 310, 311, 312, 314, 327, 332, 337, 339, 350, 353, 363, 366

Offset: 1

Eric Chen, Dec 24 2014

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

See A250177 for references.

a250179[n_] := Select[Range[n], PrimeQ@Cyclotomic[24, #] &]; a250179[366] (* Michael De Vlieger, Dec 25 2014 *)

{is(n)=isprime(polcyclo(24,n))};
for(n=1,1000, if(is(n), print1(n, ", "))) \\ G. C. Greubel, May 20 2018

A250180 Numbers n such that Phi_25(n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

1, 22, 33, 39, 43, 62, 74, 134, 142, 167, 212, 238, 287, 313, 335, 369, 414, 415, 418, 432, 509, 604, 679, 697, 770, 782, 815, 859, 874, 895, 897, 924, 1039, 1048, 1070, 1085, 1134, 1145, 1170, 1177, 1212, 1239, 1240, 1262, 1339, 1347, 1364, 1374, 1407, 1413, 1414, 1449

Offset: 1

Eric Chen, Dec 26 2014

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

See A250177 for references.

a250180[n_] := Select[Range[n], PrimeQ@Cyclotomic[25, #] &]; a250180[256]

isok(n) = isprime(polcyclo(25, n)); \\ Michel Marcus, Dec 27 2014

cp's OEIS Frontend

A100330 Positive integers k such that k^6 + k^5 + k^4 + k^3 + k^2 + k + 1 is prime.

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A250174 Numbers n such that Phi_14(n) is prime, where Phi is the cyclotomic polynomial.

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A250178 Numbers k such that Phi_22(k) is prime, where Phi is the cyclotomic polynomial.

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Magma

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A250179 Numbers n such that Phi_24(n) is prime, where Phi is the cyclotomic polynomial.

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A250180 Numbers n such that Phi_25(n) is prime, where Phi is the cyclotomic polynomial.

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Mathematica

PARI