A019320 -id:A019320 - cp's OEIS frontend

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

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

Offset: 1

Reiner Martin, Jul 04 2002

nonn

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
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

Max Alekseyev, Table of n, a(n) for n = 1..289 (terms 1..234, 235..277, and 278..289 from Yves Gallot, T. D. Noe, and Carl Pomerance, respectively)
Joerg Arndt, Matters Computational (The Fxtbook)
Yves Gallot, Cyclotomic polynomials and prime numbers
Carl Pomerance, Cyclotomic primes, arXiv:2411.04213 [math.NT], 2024.
Wikipedia, Aurifeuillean factorization
Index entries for cyclotomic polynomials, values at X

Corresponding primes are listed in A292015.

Cf. A138920-A138940.

Select[Range[600], PrimeQ[Cyclotomic[ #, 2]]&]

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

Edited by Max Alekseyev, Apr 25 2018

A085021 Number of prime factors of cyclotomic(n,2), which is A019320(n), the value of the n-th cyclotomic polynomial evaluated at x=2.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 1, 2, 3, 3, 3, 2, 3, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 3, 1, 2, 3, 2, 3, 2, 2, 3, 1, 1, 3, 1, 3, 2, 2, 2, 1, 1, 2, 2, 1, 1, 3, 4, 1, 2, 3, 2, 2, 1, 3, 4

Offset: 1

T. D. Noe, Jun 19 2003

nonn

The Mobius transform of this sequence yields A046051, the number of prime factors of Mersenne number 2^n-1.

The number of prime factors in the primitive part of 2^n-1. - T. D. Noe, Jul 19 2008

			a(11) = 2 because cyclotomic(11,2) = 2047, which has two factors: 23 and 89.

Max Alekseyev, Table of n, a(n) for n = 1..1206 (first 500 term from T. D. Noe)
Vjekoslav Kovač and Florian Luca, On the number of divisors of Mersenne numbers, arXiv:2506.04883 [math.NT], 2025. See Table 2 p. 9.
H. Mishima, Factorization of Cyclotomic Numbers
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
Eric Weisstein's World of Mathematics, Möbius Transform

omega(Phi(n,x)): this sequence (x=2), A085028 (x=3), A085029 (x=4), A085030 (x=5), A085031 (x=6), A085032 (x=7), A085033 (x=8), A085034 (x=9), A085035 (x=10).

Cf. A005420, A019320, A046051, A046801, A059499, A064078, A112927, A212953.

Join[{0}, Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 2]]][[2]], {n, 2, 100}]]

a(n) = #factor(polcyclo(n, 2))~; \\ Michel Marcus, Mar 06 2015

A063672 Sequence A019320 in binary.

Original entry on oeis.org

10, 1, 11, 111, 101, 11111, 11, 1111111, 10001, 1001001, 1011, 11111111111, 1101, 1111111111111, 101011, 10010111, 100000001, 11111111111111111, 111001, 1111111111111111111, 11001101, 100100110111, 1010101011

Offset: 0

Antti Karttunen, Aug 03 2001

Paolo Xausa, Table of n, a(n) for n = 0..900

Cf. A063697, A063699, A063671.

map(convert, A019320,binary); or up to n=104 with: map(convert,[seq(Phi_pos_terms(j,2)-Phi_neg_terms(j,2),j=0..104)],binary);

A063672[n_] := If[n == 0, 10, FromDigits[IntegerDigits[Cyclotomic[n, 2], 2]]];
Array[A063672, 30, 0] (* Paolo Xausa, Feb 26 2024 *)

A093106 Numbers k such that the k-th cyclotomic polynomial evaluated at 2 (=A019320(k)) is not coprime to k.

Original entry on oeis.org

6, 18, 20, 21, 54, 100, 110, 136, 147, 155, 156, 162, 253, 342, 486, 500, 602, 657, 812, 820, 889, 979, 1029, 1081, 1210, 1332, 1458, 2028, 2265, 2312, 2485, 2500, 2756, 3081, 3164, 3422, 3660, 3924, 4112, 4374, 4422, 4656, 4805, 5253, 5784, 5819, 6498

Offset: 1

Ralf Stephan, Mar 20 2004

nonn

Also, numbers k such that the Zsigmondy number Zs(k, 2, 1) differs from the k-th cyclotomic polynomial evaluated at 2, i.e., A064078(k) differs from A019320(k).

Numbers k > 0 such that A019320(k) is not congruent to 1 mod k. These numbers are of the form k = p^j * A002326((p-1)/2), where p is an odd prime and j > 0. Then A019320(k) mod k = gcd(A019320(k), k) = A019320(k) / A064078(k) = p. - Thomas Ordowski, Oct 07 2017

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..51350

Cf. A093107, A093108, A093109.

Select[Range[10000],GCD[#,Cyclotomic[#,2]]!=1 &] (* Emmanuel Vantieghem, Nov 13 2016 *)

isok(k) = gcd(polcyclo(k, 2), k) != 1; \\ Michel Marcus, Oct 07 2017

upto(K)=li=List();forprime(p=3,K*log(2)/log(K+1),r=znorder(Mod(2,p))*p;while(r<=K,listput(li,r);r*=p));Set(li) \\ Jeppe Stig Nielsen, Sep 10 2020

More terms from Vladeta Jovovic, Apr 03 2004
Definition corrected by Jerry Metzger, Nov 04 2009
Edited by Max Alekseyev, Oct 23 2017

A297412 Numbers k such that A019320(k) is in A217468.

Original entry on oeis.org

43, 114, 163, 258, 326, 379, 487, 758, 762, 883, 974, 978, 1459, 1766, 2274, 2647, 2918, 2922, 3079, 3943, 5294, 5298, 5419, 6158, 7886, 8754, 9199, 10838, 11827, 14407, 15882, 16759, 18398, 18474, 18523, 23654, 23658, 24967, 26407, 28814, 32514, 33518, 37046, 37339, 39367

Offset: 1

Max Alekseyev, Dec 29 2017

nonn

Set difference of A297413 and A072226.

is_A297412(n) = my(m=polcyclo(n, 2)); Mod(2, m*(m-1))^m==2 && !ispseudoprime(m);

A297413 Numbers k such that 2^m == 2 (mod m*(m-1)), where m=A019320(k).

Original entry on oeis.org

2, 3, 6, 7, 14, 19, 38, 42, 43, 86, 114, 127, 163, 254, 258, 326, 379, 487, 758, 762, 883, 974, 978, 1459, 1766, 2274, 2647, 2918, 2922, 3079, 3943, 5294, 5298, 5419, 6158, 7886, 8754, 9199, 10838, 11827, 14407, 15882, 16759, 18398, 18474, 18523, 23654, 23658, 24967, 26407, 28814, 32514, 33518, 37046, 37339, 39367

Offset: 1

Max Alekseyev, Dec 29 2017

nonn

Also, numbers k such that A019320(k) belongs to A069051 or A217468.

Cf. A069051, A217468, A297414.

Contains A297412 as a subsequence.

is_A297413(k) = my(m=polcyclo(k,2)); Mod(2,m*(m-1))^m==2;

A297414 Numbers k such that 2^m == 2 (mod m*(m+1)), where m = A019320(k).

Original entry on oeis.org

1, 4, 9, 12, 25, 36, 40, 52, 80, 92, 124, 150, 208, 306, 361, 630, 656, 1648, 1780, 2508, 3300, 3540, 5728, 6260, 6450, 7500, 10820, 12656, 14076, 14132, 18836, 20960, 23456, 24272, 35280, 43136

Offset: 1

Max Alekseyev, Dec 29 2017

nonn

Also, numbers k such that A019320(k) belongs to A216822.

Cf. A019320, A216822, A297413, A297415.

is_A297414(k) = my(m=polcyclo(k, 2)); Mod(2, m*(m+1))^m==2;

A297415 Numbers k such that A019320(k) is in A217465.

Original entry on oeis.org

25, 36, 52, 92, 124, 306, 361, 630, 656, 1648, 1780, 2508, 3300, 3540, 5728, 6260, 6450, 7500, 10820, 12656, 14076, 14132, 18836, 20960, 23456, 24272, 35280, 43136

Offset: 1

Max Alekseyev, Dec 29 2017

Cf. A019320, A217465, A297412.

Set difference of A297414 and ({1} U A072226).

is_A297415(n) = my(m=polcyclo(n, 2)); (m>1) && Mod(2, m*(m+1))^m==2 && !ispseudoprime(m);

A368424 Numbers k such that gcd(A019320(k), A019321(k)) > 1.

Original entry on oeis.org

4, 11, 18, 20, 23, 28, 35, 43, 48, 52, 83, 95, 100, 119, 131, 138, 148, 155, 162, 166, 172, 179, 191, 196, 204, 210, 214, 239, 251, 253, 268, 292, 299, 300, 316, 323, 342, 359, 371, 378, 388, 419, 431, 443, 460, 463, 491, 500, 508, 515, 537, 556, 564, 575

Offset: 1

Tomohiro Yamada, Dec 24 2023

nonn

The corresponding greatest common divisors are given in A368425.

			a(1) = 4 since A019320(4) = 5 and A019321(4) = 10.

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

Cf. A019320, A019321, A191609 (prime factors of such gcds), A368425.

select(k -> igcd(numtheory:-cyclotomic(k,2),
numtheory:-cyclotomic(k,3)) > 1, [$1..1000]); # Robert Israel, Dec 26 2023

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

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

A333973 Numbers k such that A019320(k) is greater than A064078(k) and the latter is a prime or a prime power.

Original entry on oeis.org

18, 20, 21, 54, 147, 342, 602, 889, 258121

Offset: 1

Jeppe Stig Nielsen, Sep 22 2020

The unique prime factor of A064078(k) is then a unique prime to base 2 (see A161509), but not a cyclotomic number.
Subsequence of A161508. In fact, subsequence of the set difference A161508 \ A072226.
In all known examples, A064078(k) is a prime. If A064078(k) was a prime power p^j with j>1, then p would be both a Wieferich prime (A001220) and a unique prime to base 2.
Subsequence of A093106 (the characterization of A093106 can be useful when searching for more terms).
Should this sequence be infinite?

Henri Lifchitz and Renaud Lifchitz, Phi(258121,2)/719.
Wikipedia, Unique prime, section Binary unique primes.

Cf. A019320, A064078, A093106, A072226, A144755, A161508, A161509, A247071.

for(n=1,+oo,c=polcyclo(n,2); c % n < 2 && next(); c/=(c%n); ispseudoprime(if(ispower(c,,&b),b,c))&&print1(n, ", "))

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

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A085021 Number of prime factors of cyclotomic(n,2), which is A019320(n), the value of the n-th cyclotomic polynomial evaluated at x=2.

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A063672 Sequence A019320 in binary.

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A093106 Numbers k such that the k-th cyclotomic polynomial evaluated at 2 (=A019320(k)) is not coprime to k.

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A297412 Numbers k such that A019320(k) is in A217468.

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A297413 Numbers k such that 2^m == 2 (mod m*(m-1)), where m=A019320(k).

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A297414 Numbers k such that 2^m == 2 (mod m*(m+1)), where m = A019320(k).

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A297415 Numbers k such that A019320(k) is in A217465.

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A368424 Numbers k such that gcd(A019320(k), A019321(k)) > 1.

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