A095692 -id:A095692 - cp's OEIS frontend

A049407 Numbers m such that m^3 + m + 1 is prime.

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, 71, 74, 77, 83, 87, 95, 102, 104, 108, 116, 117, 120, 126, 131, 135, 143, 144, 146, 152, 153, 155, 162, 168, 177, 179, 180, 186, 191, 207, 212, 219, 221, 225, 239, 240, 243

Offset: 1

N. J. A. Sloane

For s = 5, 8, 11, 14, 17, 20, ... (A016789(s) for s>=2), m_s = 1 + m + m^s is composite for m>1. Also for m=1, m_s = 3 is a prime for any s. Here we consider the case s=3.
If m == 1 (mod 3), m_s == 0 (mod 3) for any s and is not prime for m > 1. Thus for n > 1, a(n) !== 1 (mod 3) and this is true for any similar sequence based on another s value (A002384, A049408, A075723). - Jean-Christophe Hervé, Sep 20 2014
Corresponding primes are in A095692.

			3 is a term because 1 + 3 + 3^3 = 31 is a prime.

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

Cf. A002384 (s=2), A049408 (s=4), A075723 (s=6).

Cf. A095692 (corresponding primes).

[n: n in [0..300] | IsPrime(s) where s is 1+&+[n^i: i in [1..3 by 2]]]; // Vincenzo Librandi, Jun 27 2014

A049407:=n->`if`(isprime(n^3+n+1), n, NULL): seq(A049407(n), n=1..300); # Wesley Ivan Hurt, Nov 14 2014

Select[Range[500], PrimeQ[Total[#^Range[1, 3, 2]] + 1] &] (* Vincenzo Librandi, Jun 27 2014 *)

is(n)=isprime(n^3+n+1) \\ Charles R Greathouse IV, Nov 20 2012

from sympy import isprime
def ok(m): return isprime(m**3 + m + 1)
print([m for m in range(244) if ok(m)]) # Michael S. Branicky, Feb 17 2022

A163421 Primes of the form ((p-1)/2)^3+((p+1)/2), p are prime numbers.

Original entry on oeis.org

3, 11, 31, 131, 223, 521, 739, 3391, 5851, 9283, 24419, 27031, 59359, 68963, 85229, 110641, 148931, 157519, 175673, 328579, 405299, 571871, 857471, 1561013, 1728121, 2248223, 2460511, 3112283, 3581731, 3724031, 4741801, 5735519

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

Subsequence of A095692.

((3-1)/2)^3+((3+1)/2)=1+2=3, ((5-1)/2)^3+((5+1)/2)=8+3=11, ((7-1)/2)^3+((7+1)/2)=27+4=31,..

J. Mulder, Table of n, a(n) for n = 1..10000

Cf. A162652, A163418, A163419, A163420.

f[n_]:=((p-1)/2)^3+((p+1)/2); lst={};Do[p=Prime[n];If[PrimeQ[f[p]],AppendTo[lst,f[p]]],{n,6!}];lst

Comment from Charles R Greathouse IV, Aug 11 2009

A163422 Primes p such that A071568((p-1)/2) is also prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 31, 37, 43, 59, 61, 79, 83, 89, 97, 107, 109, 113, 139, 149, 167, 191, 233, 241, 263, 271, 293, 307, 311, 337, 359, 373, 383, 439, 443, 479, 487, 491, 523, 557, 617, 641, 647, 659, 673, 683, 701, 733, 757, 811, 829, 853, 857, 859, 877

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

Primes p such that (p-1)^3/8+(p+1)/2 is also prime, i.e., in A095692.

			p=3 is in the sequence because (3-1)^3/8+(3+1)/2=3 is prime.
p=5 is in the sequence because (5-1)^3/8+(5+1)/2=11 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A162652, A163418, A163419, A163420, A163421.

[p: p in PrimesUpTo(1000) | IsPrime((p^3-3*p^2+7*p+3) div 8)]; // Vincenzo Librandi, Apr 10 2013

f[n_]:=((n-1)/2)^3+((n+1)/2); lst={}; Do[p=Prime[n]; If[PrimeQ[f[p]], AppendTo[lst, p]], {n,6!}]; lst
Select[Prime[Range[180]], PrimeQ[(#-1)^3/8+(#+1)/2]&]  (* Harvey P. Dale, Jan 05 2011 *)

Definition rewritten by R. J. Mathar, Aug 17 2009

A165946 Primes of the form p + (p-1)^3, where p is also prime.

Original entry on oeis.org

3, 11, 223, 1741, 5851, 27031, 74131, 216061, 1061311, 1259821, 2000503, 4251691, 5832181, 13824241, 21024853, 30371641, 37933393, 49028263, 54010531, 67917721, 84028111, 123506491, 162771883, 185193571, 191103553, 216000601, 229221541, 250047631, 264609931

Offset: 1

Claudio Meller, Oct 01 2009

Generated by p = 2, 3, 7, 13, 19, 31, 43, 61, 103, 109, 127,... [R. J. Mathar, Oct 28 2009]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A095692, A163421. [R. J. Mathar, Oct 28 2009]

[a: p in PrimesInInterval(1, 1000) | IsPrime(a) where a is p + (p - 1)^3]; // Vincenzo Librandi, Oct 12 2012

Select[Table[Prime[i] + (Prime[i] - 1)^3, {i, 300}], PrimeQ] (* Harvey P. Dale, Oct 07 2009 *)
Select[Table[p + (p - 1)^3, {p, Prime[Range[300]]}], PrimeQ] (* Vincenzo Librandi, Oct 12 2012 *)

5 more terms from R. J. Mathar, Oct 28 2009

A182332 Primes of the form n^3 + n - 1.

Original entry on oeis.org

29, 67, 349, 1009, 3389, 4111, 5849, 9281, 15649, 19709, 35969, 46691, 59357, 79549, 97381, 132701, 140659, 166429, 250109, 389089, 474629, 531521, 658589, 804449, 830677, 884831, 1000099, 1092829, 1157729, 1295137, 1405039, 1520989, 1601729, 1728119, 1906747

Offset: 1

Alex Ratushnyak, Apr 25 2012

nonn

Infinite under Bunyakovsky's conjecture. - Charles R Greathouse IV, Apr 25 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A095692, A100698, A116581.

for(n=1,1e4,if(isprime(t=n^3+n-1),print1(t", "))) \\ Charles R Greathouse IV, Apr 25 2012

cp's OEIS Frontend

A049407 Numbers m such that m^3 + m + 1 is prime.

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A163421 Primes of the form ((p-1)/2)^3+((p+1)/2), p are prime numbers.

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A163422 Primes p such that A071568((p-1)/2) is also prime.

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Author

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Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Extensions

A165946 Primes of the form p + (p-1)^3, where p is also prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Extensions

A182332 Primes of the form n^3 + n - 1.

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Author

Keywords

Comments

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Crossrefs

Programs

PARI