A138940 -id:A138940 - cp's OEIS frontend

A138920 Indices k such that A020509(k)=Phi[k](-10) is prime, where Phi is a cyclotomic polynomial.

4, 5, 7, 12, 19, 24, 31, 36, 38, 46, 48, 53, 67, 75, 78, 120, 186, 196, 293, 320, 327, 369, 634, 641, 713, 770, 931, 1067, 1172, 1194, 1404, 1452, 1752, 1812, 1836, 1844, 1875, 1890, 2062, 2137, 2177, 2232, 2264, 3011, 3042, 3261, 3341, 4775, 5334, 6685

Offset: 1

M. F. Hasler, Apr 03 2008

nonn

"Unique [period] primes" (A040017) are often of the form Phi[k](10) or Phi[k](-10).

Two cyclotomic polynomial identities tightly connect this sequence to A138940: 1) Phi_2k(x) = Phi_k(-x) for odd integer k > 1. 2) Phi_4k(x) = Phi_2k(x^2) for all positive integer k. - Ray Chandler, Apr 30 2017

Ray Chandler, Table of n, a(n) for n = 1..89 (first 76 terms from Robert Price)
C. Caldwell, Unique Primes.
Index entries for cyclotomic polynomials, values at X

Cf. A019328, A040017, A138940.

Select[Range[1000], PrimeQ[Cyclotomic[#, -10]] &]

for( i=1,999, is/*pseudo*/prime( polcyclo(i,-10)) &&& print1( i",")) /* for PARI < 2.4.2 use ...subst(polcyclo(i,x),x,-10)... */

a(28)-a(43) from Robert Price, Mar 09 2012

a(44)-a(50) from Robert Price, Apr 14 2012

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

Original entry on oeis.org

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

A019328 Cyclotomic polynomials at x=10.

Original entry on oeis.org

9, 11, 111, 101, 11111, 91, 1111111, 10001, 1001001, 9091, 11111111111, 9901, 1111111111111, 909091, 90090991, 100000001, 11111111111111111, 999001, 1111111111111111111, 99009901, 900900990991, 9090909091, 11111111111111111111111, 99990001

Offset: 1

Simon Plouffe

nonn

See A138940 for indices n for which a(n) is prime. - M. F. Hasler, Jan 09 2015

Paolo Xausa, Table of n, a(n) for n = 1..900
Index entries for cyclotomic polynomials, values at X

with(numtheory,cyclotomic); f := n->subs(x=10,cyclotomic(n,x)); seq(f(i),i=0..64);

Cyclotomic[Range[30], 10] (* Paolo Xausa, Feb 26 2024 *)

A019328(n)=polcyclo(n,10) \\ Simplified by M. F. Hasler, Jan 09 2015

A138929 Twice the prime powers A000961.

Original entry on oeis.org

2, 4, 6, 8, 10, 14, 16, 18, 22, 26, 32, 34, 38, 46, 50, 54, 58, 62, 64, 74, 82, 86, 94, 98, 106, 118, 122, 128, 134, 142, 146, 158, 162, 166, 178, 194, 202, 206, 214, 218, 226, 242, 250, 254, 256, 262, 274, 278, 298, 302, 314, 326, 334, 338, 346, 358, 362, 382

Offset: 1

M. F. Hasler, Apr 04 2008

nonn

Except for the initial term a(1)=2, indices k such that A020513(k)=Phi[k](-1) is prime, where Phi is a cyclotomic polynomial.
This is illustrated by the PARI code, although it is probably more efficient to calculate a(n) as 2*A000961(n).
{ a(n)/2 ; n>1 } are also the indices for which A020500(k)=Phi[k](1) is prime.
A188666(k) = A000961(k+1) for k: a(k) <= k < a(k+1), k > 0;
A188666(a(n)) = A000961(n+1). [Reinhard Zumkeller, Apr 25 2011]

Cf. A000961, A020513, A138920-A138940, A230078 (complement).

a := n -> `if`(1>=nops(numtheory[factorset](n)),2*n,NULL):
seq(a(i),i=1..192); # Peter Luschny, Aug 12 2009

Join[{2}, Select[ Range[3, 1000], PrimeQ[ Cyclotomic[#, -1]] &]] (* Robert G. Wilson v, Mar 25 2012 - modified by Paolo Xausa, Aug 30 2024 to include a(1) *)
2*Join[{1}, Select[Range[500], PrimePowerQ]] (* Paolo Xausa, Aug 30 2024 *)

print1(2);for( i=1,999, isprime( polcyclo(i,-1)) & print1(",",i)) /* use ...subst(polcyclo(i),x,-2)... in PARI < 2.4.2. It should be more efficient to calculate a(n) as 2*A000961(n) ! */

from sympy import primepi, integer_nthroot
def A138929(n):
    def f(x): return int(n-1+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    kmin, kmax = 0,1
    while f(kmax) > kmax:
        kmax <<= 1
    while kmax-kmin > 1:
        kmid = kmax+kmin>>1
        if f(kmid) <= kmid:
            kmax = kmid
        else:
            kmin = kmid
    return kmax<<1 # Chai Wah Wu, Aug 29 2024

a(n) = 2*A000961(n).

Equals {2} union { k | Phi[k](-1)=A020513(k) is prime } = {2} union { 2k | Phi[k](1)=A020500(k) is prime }.

A138933 Indices k such that A019321(k)=Phi[k](3) is prime, where Phi is a cyclotomic polynomial.

Original entry on oeis.org

1, 3, 6, 7, 9, 10, 12, 13, 14, 15, 21, 24, 26, 33, 36, 40, 46, 60, 63, 70, 71, 72, 86, 103, 108, 130, 132, 143, 145, 154, 161, 236, 255, 261, 276, 279, 287, 304, 364, 430, 464, 513, 528, 541, 562, 665, 672, 680, 707, 718, 747, 760, 782, 828, 875, 892, 974, 984

Offset: 1

M. F. Hasler, Apr 03 2008

nonn

Robert Price, Table of n, a(n) for n = 1..167
Yves Gallot, Cyclotomic polynomials and prime numbers
Index entries for cyclotomic polynomials, values at X

Cf. A019321, A072226, A138920-A138940.

Select[Range[1000], PrimeQ[Cyclotomic[#, 3]] &]

for( i=1,999, ispseudoprime( polcyclo(i,3)) && print1( i","))

A252491 a(n) = (10^(n^2) - 1)/(10^n - 1).

Original entry on oeis.org

1, 101, 1001001, 1000100010001, 100001000010000100001, 1000001000001000001000001000001, 1000000100000010000001000000100000010000001, 100000001000000010000000100000001000000010000000100000001, 1000000001000000001000000001000000001000000001000000001000000001

Offset: 1

M. F. Hasler, Jan 08 2015

When written in base 10, the terms consist of n digits '1' separated by strings of n-1 digits '0'.
This sequence is relevant for counterexamples to a conjecture in A086766: If p is prime and a(p) is not prime, then A086766(10^(p-1)) = 0.
a(n) is the product of A019328(d) for all d that divide n^2 but not n. - Robert Israel, Jan 08 2015
If a(n) is a prime then n is a prime. What is the smallest prime term greater than 101 in this sequence? - Farideh Firoozbakht, Jan 08 2015
According to what precedes, a(n) is prime iff A019328(d) is prime, where d is the only divisor of n^2 which is not a divisor of n, i.e., iff n is a prime and n^2 is in A138940. No such term is known, except for n=2. - M. F. Hasler, Jan 09 2015

Cf. A128889 (for 2 instead of 10).

seq((10^(n^2)-1)/(10^n-1), n=1..20); # Robert Israel, Jan 08 2015

PARI
```
A252491(n)=(10^(n^2)-1)\(10^n-1)
```

A138919 Indices k such that A020510(k)=Phi[k](-11) is prime, where Phi is a cyclotomic polynomial.

Original entry on oeis.org

5, 7, 18, 20, 21, 30, 34, 36, 38, 52, 54, 60, 90, 104, 117, 123, 146, 159, 179, 182, 229, 278, 388, 405, 410, 439, 552, 557, 735, 806, 807, 1220, 1272, 1568, 1688, 1696, 1710, 1814, 2136, 2262, 2288, 2862, 3679, 3814, 4058, 4304, 4480, 5070, 5136, 5154

Offset: 1

M. F. Hasler, Apr 04 2008

nonn

Larger values are probable primes.

Robert Price, Table of n, a(n) for n = 1..91
Yves Gallot, Cyclotomic polynomials and prime numbers
Index entries for cyclotomic polynomials, values at X

Cf. A020510, A138920-A138940, A070518-A070527.

Select[Range[1000], PrimeQ[Cyclotomic[#, -11]] &]

for( i=1,999, ispseudoprime( polcyclo(i,-11)) & print1(",",i)) /* use ...subst(polcyclo(i),x,-11)... in PARI < 2.4.2 */

Edited by T. D. Noe, Oct 30 2008
a(32)-a(44) from Robert Price, Mar 16 2012
a(45)-a(50) from Robert Price, Apr 14 2012

A138934 Indices k such that A019322(k) = Phi[k](4) is prime, where Phi is a cyclotomic polynomial.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 20, 28, 40, 60, 92, 96, 104, 140, 148, 156, 300, 356, 408, 596, 612, 692, 732, 756, 800, 952, 996, 1004, 1228, 1268, 2240, 2532, 3060, 3796, 3824, 3944, 5096, 5540, 7476, 7700, 8544, 9800, 14628, 15828, 16908, 18480, 20260, 21924, 24656, 38456

Offset: 1

M. F. Hasler, Apr 03 2008

nonn

It appears that except for 1,2 and 6, all terms of this sequence are multiples of 4.
It also appears that all cyclotomic polynomials, Phi(k,x), where k is a multiple of 4 have no odd powers of x.  For example, Phi(20,x) = x^8 - x^6 + x^4 - x^2 + 1.  This implies that Phi(k,x) = Phi(k,-x), where k is a multiple of 4. - Robert Price, Apr 13 2012
Second comment is true; this follows from applying Theorem 1.1 in the Gallot paper with p = 2 and m even. - Charlie Neder, May 16 2019

Robert Price, Table of n, a(n) for n = 1..51
Yves Gallot, Cyclotomic polynomials and prime numbers
Index entries for cyclotomic polynomials, values at X

Cf. A019322, A072226, A138920-A138940.

Select[Range[1000], PrimeQ[Cyclotomic[#, 4]] &]

for( i=1,999, ispseudoprime( polcyclo(i,4)) && print1( i","))

a(29)-a(51) from Robert Price, Apr 12 2012

A138935 Indices k such that A019323(k)=Phi[k](5) is prime, where Phi is a cyclotomic polynomial.

Original entry on oeis.org

3, 7, 10, 11, 12, 13, 24, 28, 47, 48, 49, 56, 57, 88, 90, 92, 108, 110, 116, 120, 127, 134, 141, 149, 161, 181, 198, 202, 206, 236, 248, 288, 357, 384, 420, 458, 500, 530, 536, 619, 620, 694, 798, 897, 929, 981, 992, 1064, 1134, 1230, 1670, 1807, 2094, 2369

Offset: 1

M. F. Hasler, Apr 03 2008

nonn

Can there be an odd multiple of 5 in this sequence?

Robert Price, Table of n, a(n) for n = 1..110
Yves Gallot, Cyclotomic polynomials and prime numbers
Index entries for cyclotomic polynomials, values at X

Cf. A019323, A138920-A138940.

Select[Range[1000], PrimeQ[Cyclotomic[#, 5]] &]

for( i=1,999, ispseudoprime( polcyclo(i,5)) && print1( i","))

a(48)-a(54) from Robert Price, Apr 14 2012

A138936 Indices n such that A019324(k)=Phi[k](6) is prime, where Phi is a cyclotomic polynomial.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 18, 21, 22, 24, 29, 30, 42, 50, 62, 71, 86, 90, 94, 118, 124, 127, 144, 154, 192, 214, 271, 354, 360, 411, 480, 509, 558, 575, 663, 764, 814, 825, 874, 1028, 1049, 1050, 1102, 1113, 1131, 1158, 1376, 1464, 1468, 1535, 1622, 1782, 1834

Offset: 1

M. F. Hasler, Apr 03 2008

nonn

Robert Price, Table of n, a(n) for n = 1..103
Yves Gallot, Cyclotomic polynomials and prime numbers
Index entries for cyclotomic polynomials, values at X

Cf. A019324, A138920-A138940.

Select[Range[1000], PrimeQ[Cyclotomic[#, 6]] &]

for( i=1,999, ispseudoprime( polcyclo(i,6)) && print1( i",")) /* use ...subst(polcyclo(i),x,6)... in PARI < 2.4.2 */

a(41)-a(54) from Robert Price, Apr 16 2012

cp's OEIS Frontend

A138920 Indices k such that A020509(k)=Phi[k](-10) is prime, where Phi is a cyclotomic polynomial.

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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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A019328 Cyclotomic polynomials at x=10.

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A138929 Twice the prime powers A000961.

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A138933 Indices k such that A019321(k)=Phi[k](3) is prime, where Phi is a cyclotomic polynomial.

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A252491 a(n) = (10^(n^2) - 1)/(10^n - 1).

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A138919 Indices k such that A020510(k)=Phi[k](-11) is prime, where Phi is a cyclotomic polynomial.

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PARI

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A138934 Indices k such that A019322(k) = Phi[k](4) is prime, where Phi is a cyclotomic polynomial.

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