A019320 -id:A019320 - cp's OEIS frontend

A153601 Numbers representable as Phi(k,2), the k-th cyclotomic polynomial evaluated at 2, for some k>0.

1, 3, 5, 7, 11, 13, 17, 31, 43, 57, 73, 127, 151, 205, 241, 257, 331, 683, 2047, 2359, 2731, 3277, 4033, 5419, 8191, 43691, 61681, 65281, 65537, 80581, 131071, 174763, 261633, 262657, 524287, 599479, 838861, 1016801, 1082401, 1397419, 2796203

Offset: 1

Artur Jasinski, Dec 29 2008

nonn

Odd numbers in A019320.

For primes in this sequence see: A153602.

Cf. A019320, A153443, A153602.

Edited by T. D. Noe, Apr 27 2009

A292015 Primes of the form Phi(k, 2), where Phi is the cyclotomic polynomial.

Original entry on oeis.org

3, 7, 5, 31, 3, 127, 17, 73, 11, 13, 8191, 43, 151, 257, 131071, 524287, 683, 241, 2731, 262657, 331, 2147483647, 65537, 599479, 43691, 174763, 61681, 5419, 2796203, 4432676798593, 15790321, 2305843009213693951, 715827883, 145295143558111, 10052678938039

Offset: 1

Robert Price, Sep 07 2017

nonn

Robert Price, Table of n, a(n) for n = 1..89
Yves Gallot, Cyclotomic polynomials and prime numbers
Index entries for cyclotomic polynomials,values at X

Cf. A072226, A019320.

Select[Table[Cyclotomic[k,2], {k, 0, 100}], PrimeQ[#] &]

a(n) = A019320(A072226(n)). - Max Alekseyev, Apr 25 2018

A368425 The corresponding greatest common divisors to A368424(n).

Original entry on oeis.org

5, 23, 19, 5, 47, 29, 71, 431, 97, 53, 167, 191, 505, 239, 263, 139, 149, 311, 163, 499, 173, 359, 383, 197, 409, 211, 643, 479, 503, 23, 269, 293, 599, 1201, 317, 647, 19, 719, 743, 379, 389, 839, 863, 887, 461, 11113, 983, 5, 509, 1031, 4297, 557, 1129

Offset: 1

Tomohiro Yamada, Dec 24 2023

			a(2) = 23 since gcd(A019320(A368424(2)), A019321(A368424(2))) = gcd(2047, 88573) = 23.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A019320, A019321, A191609 (primes dividing some term of this sequence), A368424.

subs(1=NULL, [seq(igcd(numtheory:-cyclotomic(n,2), numtheory:-cyclotomic(n,3)),n=1..1000)]); # Robert Israel, Dec 26 2023

Select[GCD[Cyclotomic[Range[600], 2], Cyclotomic[Range[600], 3]],#>1&] (* Stefano Spezia, Dec 26 2023 *)

for(n=1,1000,m=gcd(polcyclo(n,2),polcyclo(n,3));if(m>1,print1(m,", ")))

A241039 Cyclotomic(n,2048).

Original entry on oeis.org

1, 2047, 2049, 4196353, 4194305, 17600780175361, 4192257, 73823022692637345793, 17592186044417, 73786976303428141057, 17583600302081, 1298708349570020393652962442872833, 17592181850113

Offset: 0

T. D. Noe, Apr 15 2014

nonn

Are all terms composite? At least the first 10000 terms are.

T. D. Noe, Table of n, a(n) for n = 0..300

Cf. A019320-A019331 (cyclotomic polynomials evaluated at 2..13).
Cf. A020500-A020513 (cyclotomic polynomials evaluated at 1, -2..-13, -1).
Cf. A117544 (least k such that cyclotomic(n,k) is prime).
Cf. A117545 (least k such that cyclotomic(k,n) is prime).

Mathematica
```
Table[Cyclotomic[k, 2048], {k, 0, 20}]
```

A250198 Numbers k such that the right Aurifeuillian primitive part of 2^k+1 is prime.

Original entry on oeis.org

2, 6, 10, 14, 18, 22, 30, 34, 38, 42, 54, 58, 66, 70, 90, 102, 110, 114, 126, 138, 170, 178, 242, 294, 314, 326, 350, 378, 462, 566, 646, 726, 758, 1150, 1242, 1302, 1482, 1558, 1638, 1710, 1770, 1970, 1994

Offset: 1

Eric Chen, Jan 18 2015

nonn

All terms are congruent to 2 modulo 4.
Let Phi_n(x) denote the n-th cyclotomic polynomial.
Numbers n such that Phi_{2nM(n)}(2) is prime.
Let J(n) = 2^n+1, J*(n) = the primitive part of 2^n+1, and this is Phi_{2n}(2).
Let M(n) = the Aurifeuillian M-part of 2^n+1, M(n) = 2^(n/2) + 2^((n+2)/4) + 1 for n congruent to 2 (mod 4).
Let M*(n) = GCD(M(n), J*(n)), this sequence lists all n such that M*(n) is prime.

			14 is in this sequence because the right Aurifeuillian primitive part of 2^14+1 is 29, which is prime.
26 is not in this sequence because the right Aurifeuillian primitive part of 2^26+1 is 8321, which equals 53 * 157 and is not prime.

Eric Chen, Gord Palameta, Factorization of Phi_n(2) for n up to 1280
Samuel Wagstaff, The Cunningham project
Eric W. Weisstein's World of Mathematics, Aurifeuillean Factorization.

Cf. A250197, A153443, A019320, A072226, A161508, A085601, A229768, A061443.

Select[Range[2000], Mod[#, 4] == 2 && PrimeQ[GCD[2^(#/2) + 2^((#+2)/4) + 1, Cyclotomic[2*#, 2]]] &]

isok(n) = isprime(gcd(2^(n/2) + 2^((n+2)/4) + 1, polcyclo(2*n, 2))); \\ Michel Marcus, Jan 27 2015

A250208 Ratio of the primitive part of 2^n-1 to the product of primitive prime factors of 2^n-1.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Eric Chen, Mar 02 2015

nonn

As with A178764, it can be shown that all terms are either 1 or prime.
a(2*3^n)   =  3 (n>=1).
a(4*5^n)   =  5 (n>=1).
a(3*7^n)   =  7 (n>=1).
a(10*11^n) = 11 (n>=1).
a(12*13^n) = 13 (n>=1).
a(8*17^n)  = 17 (n>=1).
a(18*19^n) = 19 (n>=1).
...
a(A014664(k)*prime(k)^n) = prime(k).
For other n (while Phi_n(2) is squarefree), a(n) = 1.
a(n) != 1 for n = {6, 18, 20, 21, 54, 100, 110, 136, 147, 155, 156, 162, ...}.
At least, a(A049093(n)) = 1. (In fact, since Phi_n(2) is not completely factored for n = 991, 1207, 1213, 1217, 1219, 1229, 1231, 1237, 1243, 1249, ..., so it is unknown whether they are squarefree or not, but it is likely that Phi_n(2) is squarefree for all n except 364 and 1755 (because it is likely 1093 and 3511 are the only two Wieferich primes), so a(991), a(1207), a(1213), ..., are likely to be 1.)

			a(11) = 1 since Phi_11(2) = (2^11-1)/(2-1) = 2047, and the primitive prime factors of 2^11-1 are 23 and 89, so a(11) = 2047/(23*89) = 1.
a(18) = 3 since Phi_18(2) = 2^6 - 2^3 + 1 = 57, and the only primitive prime factor of 2^18-1 is 19, so a(18) = 57/19 = 3.

Eric Chen, Table of n, a(n) for n = 1..990
Eric Chen, Table of n, a(n) for n = 1..1280 status (with unknown terms but all are conjectured to be 1)

Cf. A178764.

Cf. A001220, A237043, A093106, A049094, A049093, A014491, A072226, A161508.

a250208[n_] = If[n == 364, 1093, If[n == 1755, 3511, GCD[Cyclotomic[n, 2], n]]]; Table[a250208[n], {n, 0, 200}]

a(n) = if (n==364, 1093, if (n==1755, 3511, gcd(polcyclo(n, 2), n)));

isprimitive(p, n) = {for (r=1, n-1, if (((2^r-1) % p) == 0, return (0)); ); return (1); }
ppf(n) = {my(pf = factor(2^n-1)[,1]); prod(k=1,#pf, if (isprimitive(pf[k], n), pf[k], 1));}
a(n) = if (issquarefree(m=polcyclo(n,2)), gcd(m, n), m/ppf(n)); \\ Michel Marcus, Mar 06 2015

a(n) = A019320(n) / A064078(n) while Phi_n(2) is squarefree.
a(n) = GCD(Phi_n(2), n) while Phi_n(2) is squarefree.
Notice: a(364) = 1093, a(1755) = 3511. (See A001220.)

A317299 Semiprimes in A072226.

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 22, 26, 33, 34, 38, 46, 49, 62, 65, 69, 77, 85, 86, 93, 122, 129, 133, 145, 158, 202, 254, 334, 382, 398, 447, 471, 579, 626, 694, 745, 1402, 1727, 1781, 2353, 3415, 3418, 3481, 3817, 5053, 5234, 5403, 7078, 7617, 8033, 10967, 11581

Offset: 1

Jianing Song, Jan 22 2019

nonn

Semiprimes k such that A019320(k) is prime.

Numbers of the form p^2 where (2^(p^2) - 1)/(2^p - 1) is prime, or numbers of the form p*q where (2^(p*q) - 1)/((2^p - 1)*(2^q - 1)) is prime. Here p and q are necessarily primes.

			15 is a semiprime and Phi_15(2) = (2^15 - 1)/((2^3 - 1)*(2^5 - 1)) = 151 is prime, so 15 is a term. Here Phi_n is the n-th cyclotomic polynomial.
49 is a semiprime and Phi_49(2) = (2^49 - 1)/(2^7 - 1) = 4432676798593 is prime, so 49 is a term.

Jianing Song and Max Alekseyev, Table of n, a(n) for n = 1..79 (using data from A072226)

Cf. A019320, A072226.

for(k=1, 1000, if(isprime(polcyclo(k, 2))&&bigomega(k)==2,print1(k, ", ")))

cp's OEIS Frontend

A153601 Numbers representable as Phi(k,2), the k-th cyclotomic polynomial evaluated at 2, for some k>0.

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A292015 Primes of the form Phi(k, 2), where Phi is the cyclotomic polynomial.

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A368425 The corresponding greatest common divisors to A368424(n).

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A241039 Cyclotomic(n,2048).

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A250198 Numbers k such that the right Aurifeuillian primitive part of 2^k+1 is prime.

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A250208 Ratio of the primitive part of 2^n-1 to the product of primitive prime factors of 2^n-1.

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A317299 Semiprimes in A072226.

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