A002384 -id:A002384 - cp's OEIS frontend

A250175 Numbers n such that Phi_15(n) is prime, where Phi is the cyclotomic polynomial.

2, 3, 11, 17, 23, 43, 46, 52, 53, 61, 62, 78, 84, 88, 89, 92, 99, 108, 123, 124, 141, 146, 154, 156, 158, 163, 170, 171, 182, 187, 202, 217, 219, 221, 229, 233, 238, 248, 249, 253, 264, 274, 275, 278, 283, 285, 287, 291, 296, 302, 309, 314, 315, 322, 325, 342, 346, 353, 356, 366, 368, 372, 377, 380, 384, 394, 404, 406, 411, 420, 425

Offset: 1

Eric Chen, Dec 24 2014

nonn

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

Cf. A008864 (1), A006093 (2), A002384 (3), A005574 (4), A049409 (5), A055494(6), A100330 (7), A000068 (8), A153439 (9), A246392 (10), A162862(11), A246397 (12), A217070 (13), A006314 (16), A217071 (17), A164989(18), A217072 (19), A217073 (23), A153440 (27), A217074 (29), A217075(31), A006313 (32), A097475 (36), A217076 (37), A217077 (41), A217078(43), A217079 (47), A217080 (53), A217081 (59), A217082 (61), A006315(64), A217083 (67), A217084 (71), A217085 (73), A217086 (79), A153441(81), A217087 (83), A217088 (89), A217089 (97), A006316 (128), A153442(243), A056994 (256), A056995 (512), A057465 (1024), A057002 (2048), A088361 (4096), A088362 (8192), A226528 (16384), A226529 (32768), A226530(65536).

Select[Range[600], PrimeQ[Cyclotomic[15, #]] &] (* Vincenzo Librandi, Jan 16 2015 *)

isok(n) = isprime(polcyclo(15, n)); \\ Michel Marcus, Jan 16 2015

More terms from Vincenzo Librandi, Jan 16 2015

A250176 Numbers n such that Phi_20(n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

4, 9, 11, 16, 19, 26, 34, 45, 54, 70, 86, 91, 96, 101, 105, 109, 110, 119, 120, 126, 129, 139, 141, 149, 171, 181, 190, 195, 215, 229, 260, 276, 299, 305, 309, 311, 314, 319, 334, 339, 369, 375, 414, 420, 425, 444, 470, 479, 485, 506, 519, 534, 540, 550

Offset: 1

Eric Chen, Dec 24 2014

nonn

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

Cf. A008864 (1), A006093 (2), A002384 (3), A005574 (4), A049409 (5), A055494(6), A100330 (7), A000068 (8), A153439 (9), A246392 (10), A162862(11), A246397 (12), A217070 (13), A006314 (16), A217071 (17), A164989(18), A217072 (19), A217073 (23), A153440 (27), A217074 (29), A217075(31), A006313 (32), A097475 (36), A217076 (37), A217077 (41), A217078(43), A217079 (47), A217080 (53), A217081 (59), A217082 (61), A006315(64), A217083 (67), A217084 (71), A217085 (73), A217086 (79), A153441(81), A217087 (83), A217088 (89), A217089 (97), A006316 (128), A153442(243), A056994 (256), A056995 (512), A057465 (1024), A057002 (2048), A088361 (4096), A088362 (8192), A226528 (16384), A226529 (32768), A226530(65536).

Select[Range[600], PrimeQ[Cyclotomic[20, #]] &] (* Vincenzo Librandi, Jan 16 2015 *)

isok(n) = isprime(polcyclo(20, n)); \\ Michel Marcus, Sep 29 2015

More terms from Vincenzo Librandi, Jan 16 2015

A253240 Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.

Original entry on oeis.org

1, 1, -1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 3, 3, 1, 1, 3, 4, 7, 2, 1, 1, 4, 5, 13, 5, 5, 1, 1, 5, 6, 21, 10, 31, 1, 1, 1, 6, 7, 31, 17, 121, 3, 7, 1, 1, 7, 8, 43, 26, 341, 7, 127, 2, 1, 1, 8, 9, 57, 37, 781, 13, 1093, 17, 3, 1, 1, 9, 10, 73, 50, 1555, 21, 5461, 82, 73, 1, 1, 1, 10, 11, 91, 65, 2801, 31, 19531, 257, 757, 11, 11, 1, 1, 11, 12, 111, 82, 4681, 43, 55987, 626, 4161, 61, 2047, 1, 1

Offset: 0

Eric Chen, Apr 22 2015

Outside of rows 0, 1, 2 and columns 0, 1, only terms of A206942 occur.
Conjecture: There are infinitely many primes in every row (except row 0) and every column (except column 0), the indices of the first prime in n-th row and n-th column are listed in A117544 and A117545. (See A206864 for all the primes apart from row 0, 1, 2 and column 0, 1.)
Another conjecture: Except row 0, 1, 2 and column 0, 1, the only perfect powers in this table are 121 (=Phi_5(3)) and 343 (=Phi_3(18)=Phi_6(19)).

			Read by antidiagonals:
m\n  0   1   2   3   4   5   6   7   8   9  10  11  12
------------------------------------------------------
0    1   1   1   1   1   1   1   1   1   1   1   1   1
1   -1   0   1   2   3   4   5   6   7   8   9  10  11
2    1   2   3   4   5   6   7   8   9  10  11  12  13
3    1   3   7  13  21  31  43  57  73  91 111 133 157
4    1   2   5  10  17  26  37  50  65  82 101 122 145
5    1   5  31 121 341 781 ... ... ... ... ... ... ...
6    1   1   3   7  13  21  31  43  57  73  91 111 133
etc.
The cyclotomic polynomials are:
n        n-th cyclotomic polynomial
0        1
1        x-1
2        x+1
3        x^2+x+1
4        x^2+1
5        x^4+x^3+x^2+x+1
6        x^2-x+1
...

Eric Chen, Table of n, a(n) for n = 0..5049 (first 100 antidiagonals)
Eric Weisstein's World of Mathematics, Cyclotomic polynomial
Wikipedia, Cyclotomic polynomial
Index entries for Cyclotomic polynomials, values at X

Rows 0-16 are A000012, A023443, A000027, A002061, A002522, A053699, A002061, A053716, A002523, A060883, A060884, A060885, A060886, A060887, A060888, A060889, A060890.
Columns 0-13 are A158388, A020500, A019320, A019321, A019322, A019323, A019324, A019325, A019326, A019327, A019328, A019329, A019330, A019331.
Main diagonal is A070518.
Indices of primes in n-th row for n = 1-20 are A008864, A006093, A002384, A005574, A049409, A055494, A100330, A000068, A153439, A246392, A162862, A246397, A217070, A250174, A250175, A006314, A217071, A164989, A217072, A250176.
Indices of primes in n-th column for n = 1-10 are A246655, A072226, A138933, A138934, A138935, A138936, A138937, A138938, A138939, A138940.
Indices of primes in main diagonal is A070519.
Cf. A117544 (indices of first prime in n-th row), A085398 (indices of first prime in n-th row apart from column 1), A117545 (indices of first prime in n-th column).
Cf. A206942 (all terms (sorted) for rows>2 and columns>1).
Cf. A206864 (all primes (sorted) for rows>2 and columns>1).

Table[Cyclotomic[m, k-m], {k, 0, 49}, {m, 0, k}]

t1(n)=n-binomial(floor(1/2+sqrt(2+2*n)), 2)
t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)
T(m, n) = if(m==0, 1, polcyclo(m, n))
a(n) = T(t1(n), t2(n))

T(m, n) = Phi_m(n)

A258775 Numbers n such that 1 + sigma(n)+ sigma(n)^2 is prime.

Original entry on oeis.org

1, 2, 5, 6, 7, 8, 11, 13, 14, 15, 19, 23, 34, 37, 40, 45, 49, 53, 57, 58, 60, 61, 78, 79, 89, 92, 105, 106, 109, 123, 129, 132, 137, 138, 140, 141, 143, 148, 149, 154, 155, 156, 160, 161, 163, 165, 167, 182, 188, 191, 193, 195, 201, 208, 212, 213, 222, 226

Offset: 1

Robert Price, Jun 09 2015

nonn

Also numbers n such that A000203(n) is in A002384. - Robert Israel, Jun 09 2015

Robert Price, Table of n, a(n) for n = 1..2122
OEIS Wiki, Cyclotomic Polynomials at x=n, n! and sigma(n)

Cf. A002384, A000203, A258774, A258776.

[n: n in [1..250] | IsPrime(1 + SumOfDivisors(n)+ SumOfDivisors(n)^2)]; // Vincenzo Librandi, Jun 10 2015

select(isprime @ (t -> 1+t+t^2) @ numtheory:-sigma, [$1..1000]); # Robert Israel, Jun 09 2015

Select[ Range[10000], PrimeQ[ 1 + DivisorSigma[1, #] + DivisorSigma[1, #]^2] & ]
Select[ Range[10000], PrimeQ[ Cyclotomic[3, DivisorSigma[1, #]]] &]

for(n=1,10^3,if(isprime(1+sigma(n)+sigma(n)^2),print1(n,", "))) \\ Derek Orr, Jun 09 2015

A342690 Prime powers q in A246655 such that q^2 + q + 1 is prime.

Original entry on oeis.org

2, 3, 5, 8, 17, 27, 41, 59, 71, 89, 101, 131, 167, 173, 293, 383, 512, 677, 701, 743, 761, 773, 827, 839, 857, 911, 1091, 1097, 1163, 1181, 1193, 1217, 1331, 1373, 1427, 1487, 1559, 1583, 1709, 1811, 1847, 1931, 1973, 2129, 2273, 2309, 2339, 2411, 2663

Offset: 1

Martin Becker, May 18 2021

nonn

Also, prime powers q = p^(3^k) with prime p and nonnegative integer k and the property that q^2 + q + 1 is prime, since the exponent must be a power of 3, from the theory of cyclotomic polynomials. 17^(3^7) is in the sequence, generating a 5382-digit prime.

			5 = 5^1 is a term: 5^2 + 5 + 1 = 31 is prime.
8 = 2^3 is a term: 8^2 + 8 + 1 = 73 is prime.

Martin Becker, Table of n, a(n) for n = 1..20000

Intersection of A246655 and A002384.

Cf. A342691, A344448.

Select[Range@2000,PrimePowerQ@#&&PrimeQ[#^2+#+1]&] (* Giorgos Kalogeropoulos, May 18 2021 *)

N=50; i=0; a=vector(N); for(q=2, oo, if(isprimepower(q) && isprime(q^2+q+1), i+=1; a[i]=q; if(i==N, break))); a

A002640 Numbers k such that (k^2 + k + 1)/3 is prime.

Original entry on oeis.org

4, 7, 10, 13, 19, 28, 31, 34, 40, 43, 52, 70, 73, 76, 82, 85, 91, 97, 103, 112, 115, 124, 127, 136, 145, 148, 157, 166, 175, 187, 190, 199, 202, 223, 241, 244, 259, 265, 271, 274, 280, 286, 316, 325, 358, 370, 376, 385, 388, 409, 421, 427, 442, 460, 469, 472

Offset: 1

N. J. A. Sloane

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 1, pp. 245-259.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]
Hajrudin Fejzić, Dan Rinne, and Bob Stein, On Sums of Cubes, The College Mathematics Journal, Vol. 36, No. 3 (May, 2005), pp. 226-228. See p. 228.

Cf. A002384.

[n: n in [4..500] | IsPrime((n^2+n+1) div 3)]; // Vincenzo Librandi, Nov 18 2010

Select[Range[500], PrimeQ[(#^2 + # + 1)/3] &] (* Vincenzo Librandi, Sep 25 2012 *)

isok(k) = my(x=k^2+k+1); !(x%3) && isprime(x/3); \\ Michel Marcus, Aug 22 2025

A107317 Semiprimes of the form 2*(m^2 + m + 1) (implying that m^2 + m + 1 is a prime).

Original entry on oeis.org

6, 14, 26, 62, 86, 146, 314, 422, 482, 614, 842, 926, 1202, 1514, 2246, 2966, 3446, 5102, 5942, 6614, 7082, 7814, 8846, 9662, 10226, 11402, 12014, 12326, 12962, 16022, 16382, 19802, 20606, 22262, 24422, 24866, 27614, 28562, 34586, 38366, 40046

Offset: 1

Giovanni Teofilatto, May 21 2005

Twice A002383.

Also semiprimes n such that 2*n - 3 is a square. - Giovanni Teofilatto, Dec 29 2005. This coincidence was noticed by Andrew S. Plewe. Proof that this is the same sequence: If X is n^2+(n+1)^2+1, then 2X-3 is 4n^2+4n+1 = (2n+1)^2. And if 2X-3 is a square, then since it's odd, 2X-3 = (2n+1)^2 and X = n^2+(n+1)^2+1. - Don Reble, Apr 18 2007

			a(1)=6 because 1^2 + 2^2 + 1 = 6 = 2*3;
a(2)=14 because 2^2 + 3^2 + 1 = 14 = 2*7;
a(3)=26 because 3^2 + 4^2 + 1 = 26 = 13*2.

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Cf. A002383, A002384.

2(#^2 + # + 1) & /@ Select[ Range[144], PrimeQ[ #^2 + # + 1] &] (* Robert G. Wilson v, May 28 2005 *)
fQ[n_] := Plus @@ Last /@ FactorInteger@n == 2 && IntegerQ@Sqrt[2n - 3]; Select[ Range@43513, fQ[ # ] &] (* Robert G. Wilson v *)

for(n=2,100000,if(bigomega(n)==2&&issquare(2*n-3),print1(n,","))) /* Lambert Herrgesell */

a(n) = 2*A002383(n).

a(n) = 2*(A002384(n)^2+A002384(n)+1).

Edited by Robert G. Wilson v, May 28 2005

Re-edited by N. J. A. Sloane, Apr 18 2007

A174967 Smallest number of the form k^2 + k + 1 with n distinct prime divisors.

Original entry on oeis.org

1, 3, 21, 273, 10101, 316407, 6914271, 2424626841, 346084535811, 6177672967557, 1741866776384007, 92264158181274807, 103008522046409631057, 22810816825458528984663, 2220066397007943013450011, 545889722100356705628041121, 73293936170018923619553695493

Offset: 0

Michel Lagneau, Apr 02 2010

nonn

If k == 2 (mod 3), all prime divisors of k^2 + k + 1 are congruent to 1 (mod 3), and if k == 1 (mod 3), the number 3 is divisor, and the other divisors are congruent to 1 (mod 3).
Proof: first case: k == 2 (mod 3): let q divide k^2 + k + 1. Then 4q divides 4*(k^2 + k + 1) = (2k+1)^2 + 3, and (-3/q)=1, where (a/b) is the Legendre symbol. By using the law of quadratic reciprocity, we obtain (-3/q) = (-1/q)(3/q) = (-1/q)(q/3)(-1)^((q-1)/2)(3-1)/2)) = ((-1)^(q-1)/2)((-1)^(q-1)/2)(q/3) = (q/3). Suppose q !== 1 (mod 3). Then k^2 + k + 1 !== 0 (mod 3) => q == 2 (mod 3), and then (q/3) = -1 => (-3/q) = -1, a contradiction. So q == 1 (mod 3).
Second case: k == 1 (mod 3) => 3 is divisor of k^2 + k + 1, and the other divisors q == 1 (mod 3).
a(11) <= 4943071434145592163, a(12) <= 2702887058650660754061, a(13) <= 896265629366361887178273, a(14) <= 72053193593257327979705541. - Michael S. Branicky, Mar 21 2021
Is a(n) squarefree? The first 16 terms are. - David A. Corneth, Mar 21 2021

			21 = 3*7;
273 = 3*7*13;
10101 = 3*7*13*37;
316407 = 3*7*13*19*61;
6914271 = 3*7*13*19*31*43;
2424626841 = 3*7*13*19*61*79*97;
346084535811 = 3*7*19*37*43*67*79*103;
6177672967557 = 3*7*13*19*31*43*61*97*151;
1741866776384007 = 3*7*13*19*31*37*43*67*151*673.

L. Poletti, Le serie dei numeri primi appartenente alle due forme quadratiche (A) n^2+n+1 e (B) n^2+n-1 per l'intervallo compreso entro 121 milioni, e cioè per tutti i valori di n fino a 11000, Atti della Reale Accademia Nazionale dei Lincei, Memorie della Classe di Scienze Fisiche, Matematiche e Naturali, s. 6, v. 3 (1929), pages 193-218.

Cf. A002384, A002383.

A174967 := proc(n)
        for k from 1 do
                a := k^2+k+1 ;
                if A001221(a) = n then
                        return a;
                end if;
        end do:
end proc: # R. J. Mathar, Jul 06 2012

from sympy import primefactors
def a(n):
  k = 1
  while len(primefactors(k**2 + k + 1)) != n: k += 1
  return k**2 + k + 1
print([a(n) for n in range(1, 8)]) # Michael S. Branicky, Mar 21 2021

a(10) from Michael S. Branicky, Mar 21 2021

a(0), a(11)-a(16) from David A. Corneth, Mar 21 2021

A236056 Numbers k such that k^2 +- k +- 1 is prime for all four possibilities.

Original entry on oeis.org

3, 6, 21, 456, 1365, 2205, 2451, 2730, 8541, 18486, 32199, 32319, 32781, 45864, 61215, 72555, 72561, 82146, 83259, 86604, 91371, 95199, 125334, 149331, 176889, 182910, 185535, 210846, 225666, 226254, 288420, 343161, 350091, 403941, 411501, 510399, 567204

Offset: 1

Derek Orr, Jan 18 2014

nonn

The only prime in this sequence is a(1) = 3.

			1365^2 + 1365 + 1 = 1864591,
1365^2 + 1365 - 1 = 1864589,
1365^2 - 1365 + 1 = 1861861, and
1365^2 - 1365 - 1 = 1861859 are all prime, so 1365 is a term of this sequence.

Numbers in the intersection of A002384, A045546, A055494, and A002328.

Numbers in the intersection of A131530 and A088485.

q:= k-> andmap(isprime, [seq(seq(k^2+i+j, j=[k, -k]), i=[1, -1])]):
select(q, [3*t$t=1..200000])[];  # Alois P. Heinz, Feb 25 2020

Select[Range[568000],AllTrue[Flatten[{#^2+#+{1,-1},#^2-#+{1,-1}},1],PrimeQ]&] (* Harvey P. Dale, Jul 31 2022 *)

import sympy
from sympy import isprime
{print(p) for p in range(10**6) if isprime(p**2+p+1) and isprime(p**2-p+1) and isprime(p**2+p-1) and isprime(p**2-p-1)}

A067664 Numbers n such that n^2 + 1 and n^2 + n + 1 are primes.

Original entry on oeis.org

1, 2, 6, 14, 20, 24, 54, 66, 90, 110, 150, 176, 206, 236, 314, 584, 644, 686, 696, 860, 864, 890, 920, 950, 960, 1070, 1146, 1274, 1314, 1340, 1434, 1440, 1494, 1566, 1616, 1644, 1676, 1700, 1716, 1970, 1974, 2054, 2064, 2136, 2360, 2430, 2456, 2604, 2646

Offset: 1

Benoit Cloitre, Feb 23 2002

All the terms are even numbers except for a(1) = 1. - Metin Sariyar, Nov 23 2019

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A005574, A002384.

[n: n in [0..10000]| IsPrime(n^2+1) and IsPrime(n^2+n+1)] // Vincenzo Librandi, Aug 07 2010

Join[{1}, Select[Range[2, 10^6, 2], PrimeQ[#^2+1]&&PrimeQ[#^2+#+1]&]] (* Metin Sariyar, Nov 23 2019 *)

cp's OEIS Frontend

A250175 Numbers n such that Phi_15(n) is prime, where Phi is the cyclotomic polynomial.

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A250176 Numbers n such that Phi_20(n) is prime, where Phi is the cyclotomic polynomial.

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A253240 Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.

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A258775 Numbers n such that 1 + sigma(n)+ sigma(n)^2 is prime.

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A342690 Prime powers q in A246655 such that q^2 + q + 1 is prime.

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A002640 Numbers k such that (k^2 + k + 1)/3 is prime.

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Magma

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A107317 Semiprimes of the form 2*(m^2 + m + 1) (implying that m^2 + m + 1 is a prime).

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