A049407 -id:A049407 - cp's OEIS frontend

A075725 Duplicate of A049407.

1, 2, 3, 5, 6, 8, 9, 12, 15, 17, 18, 21, 29, 30, 32, 39, 41, 42, 44, 48, 53, 54, 56, 60, 69

Offset: 1

dead

A002384 Numbers m such that m^2 + m + 1 is prime.

1, 2, 3, 5, 6, 8, 12, 14, 15, 17, 20, 21, 24, 27, 33, 38, 41, 50, 54, 57, 59, 62, 66, 69, 71, 75, 77, 78, 80, 89, 90, 99, 101, 105, 110, 111, 117, 119, 131, 138, 141, 143, 147, 150, 153, 155, 161, 162, 164, 167, 168, 173, 176, 188, 189, 192, 194, 203, 206, 209, 215

Offset: 1

N. J. A. Sloane

A002383 lists the corresponding primes. - Bernard Schott, Dec 22 2012
This is also the list of bases where 111 represents a prime number. - Christian N. K. Anderson, Mar 28 2013
If d>1 divides m^2 + m + 1, then m + k*d is not in the sequence, for all k>=1. - Gionata Neri, Mar 04 2017

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 1, pp. 245-259.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 46.
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.
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).

Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]
Pantelis A. Damianou, On prime values of cyclotomic polynomials, arXiv:1101.1152 [math.NT], Jan 06 2011.
Oliver Knill, Goldbach for Gaussian, Hurwitz, Octavian and Eisenstein primes, arXiv preprint arXiv:1606.05958 [math.NT], 2016.
Oliver Knill, Some experiments in number theory, arXiv preprint arXiv:1606.05971 [math.NT], 2016.

Cf. A002383, A049407, A049408, A075723, A088503, A110284.

[ n: n in [1..300] | IsPrime(n^2+n+1)] // Vincenzo Librandi, Nov 21 2010

Select[Range@ 216, PrimeQ[#^2 + # + 1] &] (* Michael De Vlieger, Mar 06 2017 *)

is(n)=isprime(n^2+n+1) \\ Charles R Greathouse IV, Jan 21 2014

a(n) = (A088503(n) - 1)/2. - Ray Chandler

Extended by Ray Chandler, Sep 07 2005

A244376 Numbers k such that 1 + k + k^3 + k^5 + k^7 + k^9 + k^11 is prime.

Original entry on oeis.org

1, 2, 10, 40, 47, 55, 62, 121, 137, 152, 167, 201, 233, 278, 290, 293, 313, 333, 370, 382, 430, 452, 460, 506, 546, 555, 613, 625, 642, 675, 705, 711, 752, 767, 793, 797, 831, 835, 837, 872, 878, 891, 906, 917, 923, 978, 985, 1005, 1012, 1017, 1018, 1021

Offset: 1

Vincenzo Librandi, Jun 27 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1400

Cf. A127936.

Cf. numbers n such that 1+n+n^3 + ... + n^k, with k odd: A006093 (k=1), A049407 (k=3), A124154 (k=5), A124150 (k=7), A124163 (k=9), this sequence (k=11), A124164 (k=13), A244377 (k=15), A244378 (k=17), A124178 (k=19), A244379 (k=21), A124181 (k=23), A244380 (k=25), A124185 (k=27), A244383 (k=29), A124186 (k=31), A244384 (k=33), A124187 (k=35), A244385 (k=37), A124189 (k=39), A244386 (k=41), A124200 (k=43), A244387 (k=45), A124205 (k=47), A244388 (k=49), A124206 (k=51), A244389 (k=53), A124207 (k=55), A244390 (k=57), A124208 (k=59), A244391 (k=61), A124209 (k=63).

[n: n in [0..1500] | IsPrime(s) where s is 1+&+[n^i: i in [1..11 by 2]]];

Select[Range[4000], PrimeQ[Total[#^Range[1, 11, 2]] + 1] &]

isok(n) = isprime(1 + n + n^3 + n^5 + n^7 + n^9 + n^11); \\ Michel Marcus, Jun 27 2014

i,n = var('i,n')
[n for n in (1..2000) if is_prime(1+(n^(2*i+1)).sum(i,0,5))] # Bruno Berselli, Jun 27 2014

A124178 Numbers n such that 1 + n + n^3 + n^5 + n^7 + n^9 + n^11 + n^13 + n^15 + n^17 + n^19 is prime.

Original entry on oeis.org

1, 3, 6, 33, 36, 61, 70, 99, 168, 229, 267, 268, 321, 325, 337, 366, 387, 448, 456, 457, 498, 513, 532, 546, 591, 621, 624, 637, 835, 858, 910, 927, 961, 981, 1045, 1125, 1213, 1237, 1242, 1257, 1341, 1357, 1437, 1458, 1461, 1462, 1482, 1491, 1572, 1579, 1581

Offset: 1

Artur Jasinski, Dec 13 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..1800

Cf. A049407, similar sequences listed in A244376.

[n: n in [0..2000] | IsPrime(1 +n +n^3 +n^5 +n^7 +n^9 +n^11 +n^13 +n^15 +n^17 +n^19)]; // Vincenzo Librandi, Nov 12 2010

[n: n in [0..2000] | IsPrime(s) where s is 1+&+[n^i: i in [1..19 by 2]]]; // Vincenzo Librandi, Jun 28 2014

Do[If[PrimeQ[1 + n + n^3 + n^5 + n^7 + n^9 + n^11 + n^13 + n^15 + n^17 + n^19], Print[n]], {n, 1, 1000}]
Select[Range[3000], PrimeQ[Total[#^Range[1, 19, 2]] + 1] &] (* Vincenzo Librandi, Jun 28 2014 *)

is(n)=n==1 || isprime((n^21-n)/(n^2-1)+1) \\ Charles R Greathouse IV, Jul 02 2013

i,n = var('i,n')
[n for n in (1..2000) if is_prime(1+(n^(2*i+1)).sum(i,0,9))] # Bruno Berselli, Jun 28 2014

A124181 Numbers n such that 1 + n + n^3 + n^5 + n^7 + n^9 + n^11 + n^13 + n^15 + n^17 + n^19 + n^21 + n^23 is prime.

Original entry on oeis.org

1, 3, 69, 86, 104, 110, 138, 146, 210, 238, 247, 260, 264, 269, 316, 436, 572, 600, 621, 654, 666, 715, 737, 740, 744, 754, 779, 1056, 1156, 1159, 1216, 1218, 1221, 1343, 1419, 1434, 1442, 1524, 1580, 1603, 1676, 1680, 1731, 1742, 1804, 1952, 1956, 1985

Offset: 1

Artur Jasinski, Dec 13 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..1390

Cf. A049407, similar sequences listed in A244376.

[n: n in [0..2000] | IsPrime(s) where s is 1+&+[n^i: i in [1..23 by 2]]]; // Vincenzo Librandi, Jun 28 2014

Do[If[PrimeQ[1 + n + n^3 + n^5 + n^7 + n^9 + n^11 + n^13 + n^15 + n^17 + n^19 + n^21 + n^23], Print[n]], {n, 1, 1400}]
Select[Range[3000], PrimeQ[Total[#^Range[1, 23, 2]] + 1] &] (* Vincenzo Librandi, Jun 28 2014 *)

is(n)=n==1 || isprime((n^25-n)/(n^2-1)+1) \\ Charles R Greathouse IV, Jul 02 2013

i,n = var('i,n')
[n for n in (1..2000) if is_prime(1+(n^(2*i+1)).sum(i,0,11))] # Bruno Berselli, Jun 27 2014

1 together with numbers n such that (n^25-n)/(n^2-1) + 1 is prime. - Charles R Greathouse IV, Jul 02 2013

A049408 Numbers k such that k^4 + k + 1 is prime.

Original entry on oeis.org

1, 2, 5, 6, 9, 11, 12, 14, 24, 26, 32, 36, 44, 47, 60, 69, 72, 74, 77, 89, 90, 102, 107, 119, 126, 131, 146, 147, 159, 162, 170, 171, 186, 191, 197, 204, 206, 219, 239, 240, 252, 266, 284, 285, 290, 296, 300, 324, 347, 351, 362, 384, 426, 437, 459, 465, 470

Offset: 1

N. J. A. Sloane

For s = 5,8,11,14,17,20,..., n_s = 1 + n + n^s is always composite for any n > 1. Also for n=1, n_s=3 is a prime for any s. Here we consider the case s=4.

			26 is a term because at s=4, n=26, n_s = 1 + n + n^s = 457003 is a prime.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A002384, A075723, A049407.

[n: n in [0..1000] | IsPrime(s) where s is 1+n+n^4]; // Vincenzo Librandi, Jul 28 2014

Select[Range[1000], PrimeQ[1 + # + #^4] &] (* Vincenzo Librandi, Jul 28 2014 *)

for(n=1, 1000, if(isprime(1+n+n^4), print1(n",")))

A126908 Numbers k such that 1 + k^2 + k^4 + k^6 + k^7 is prime.

Original entry on oeis.org

1, 4, 13, 15, 24, 30, 37, 40, 55, 93, 138, 139, 148, 153, 154, 159, 160, 165, 184, 195, 204, 223, 258, 303, 355, 360, 373, 459, 472, 475, 510, 519, 534, 577, 594, 607, 615, 627, 658, 672, 688, 723, 735, 739, 795, 805, 807, 817, 819, 820, 847, 874, 879, 904

Offset: 1

Artur Jasinski, Dec 31 2006

nonn

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A049407, A124175 A124176 A124177, A124178, A124179, A124180, A124181, A126908-A126916.

a = {}; Do[If[PrimeQ[1 + n^2 + n^4 + n^6 + n^7], AppendTo[a, n]], {n, 1, 1400}]; a
Select[Range[1000],PrimeQ[1+#^2+#^4+#^6+#^7]&] (* Harvey P. Dale, Jan 15 2016 *)

is(n)=isprime(1+n^2+n^4+n^6+n^7) \\ Charles R Greathouse IV, Feb 17 2017

A126916 Numbers n such that 1 + n^2 + n^4 + n^6 + n^8 + n^10 + n^12 + n^14 + n^16 + n^18 + n^20 + n^22 + n^23 is prime.

Original entry on oeis.org

1, 2, 11, 23, 47, 64, 77, 80, 103, 251, 290, 321, 331, 335, 375, 382, 387, 403, 507, 568, 590, 594, 649, 801, 805, 828, 830, 840, 847, 854, 905, 925, 926, 959, 982, 986, 1034, 1086, 1094, 1102, 1122, 1129, 1147, 1160, 1391

Offset: 1

Artur Jasinski, Dec 31 2006

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A049407, A124175 A124176 A124177, A124178, A124179, A124180, A124181, A126908-A126916.

a = {}; Do[If[PrimeQ[1 + n^2 + n^4 + n^6 + n^8 + n^10 + n^12 + n^14 + n^16 + n^18 + n^20 + n^22 + n^23], AppendTo[a, n]], {n, 1, 1400}]; a
Select[Range[1400],PrimeQ[Total[#^Range[2,22,2]]+1+#^23]&] (* Harvey P. Dale, Oct 04 2018 *)

is(n)=isprime(1+n^2+n^4+n^6+n^8+n^10+n^12+n^14+n^16+n^18+n^20+n^22+n^23) \\ Charles R Greathouse IV, Feb 17 2017

A236477 Numbers k such that k^3 - k + 1 is prime.

Original entry on oeis.org

2, 4, 6, 7, 10, 11, 14, 15, 19, 21, 22, 25, 26, 31, 35, 39, 41, 42, 45, 50, 52, 54, 57, 62, 75, 77, 84, 85, 87, 90, 92, 95, 99, 101, 102, 106, 111, 116, 120, 125, 129, 130, 132, 134, 136, 139, 140, 141, 147, 155, 167, 169, 176, 189, 195, 202, 221, 230, 237

Offset: 1

Derek Orr, Jan 26 2014

nonn

			26^3 - 26 + 1 = 17551 is prime. So 26 is a member of this sequence.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A049407, A126421, A236478.

Select[Range[250],PrimeQ[#^3-#+1]&] (* Harvey P. Dale, Dec 06 2017 *)

s=[]; for(n=1, 500, if(isprime(n^3-n+1), s=concat(s, n))); s \\ Colin Barker, Jan 27 2014

import sympy
from sympy import isprime
{print(n) for n in range(10**3) if isprime(n**3-n+1)}

More terms from Colin Barker, Jan 27 2014

A075723 Numbers n such that 1 + n + n^6 is a prime.

Original entry on oeis.org

1, 2, 3, 6, 8, 15, 17, 29, 30, 32, 45, 48, 59, 72, 74, 80, 87, 128, 141, 153, 155, 156, 158, 176, 182, 191, 197, 210, 216, 230, 273, 284, 293, 305, 314, 356, 366, 380, 384, 399, 402, 407, 408, 410, 413, 420, 435, 443, 447, 450, 473, 479, 497

Offset: 1

Zak Seidov, Oct 03 2002

For s = 5,8,11,14,17,20,..., n_s=1+n+n^s is always composite for any n>1. Also at n=1, n_s=3 is a prime for any s. So it is interesting to consider only the cases of s != 5,8,11,14,17,20,... and n>1. Here I consider the case s=6 and find several first n's making n_s a prime (or a probable prime).

			15 is OK because at s=6, n=15, n_s = 1 + n + n^s = 11390641 is a prime.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A002384, A049407, A049408.

[n: n in [0..500] | IsPrime(s) where s is 1+n+n^6]; // Vincenzo Librandi, Jul 28 2014

Select[Range[500], PrimeQ[1 + # + #^6] &] (* Vincenzo Librandi, Jul 28 2014 *)

for(n=1,10^3,if(isprime(n^6+n+1),print1(n,", "))) \\ Derek Orr, Feb 07 2015

cp's OEIS Frontend

A075725 Duplicate of A049407.

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A002384 Numbers m such that m^2 + m + 1 is prime.

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A244376 Numbers k such that 1 + k + k^3 + k^5 + k^7 + k^9 + k^11 is prime.

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A124178 Numbers n such that 1 + n + n^3 + n^5 + n^7 + n^9 + n^11 + n^13 + n^15 + n^17 + n^19 is prime.

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A124181 Numbers n such that 1 + n + n^3 + n^5 + n^7 + n^9 + n^11 + n^13 + n^15 + n^17 + n^19 + n^21 + n^23 is prime.

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A049408 Numbers k such that k^4 + k + 1 is prime.

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A126908 Numbers k such that 1 + k^2 + k^4 + k^6 + k^7 is prime.

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A126916 Numbers n such that 1 + n^2 + n^4 + n^6 + n^8 + n^10 + n^12 + n^14 + n^16 + n^18 + n^20 + n^22 + n^23 is prime.

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