A162291 -id:A162291 - cp's OEIS frontend

A162293 Numbers k such that k^2*(k-1)-1 is prime.

2, 3, 4, 6, 7, 9, 12, 13, 18, 21, 22, 30, 33, 46, 48, 57, 58, 61, 66, 67, 75, 79, 85, 87, 90, 94, 96, 99, 100, 106, 111, 114, 117, 118, 120, 121, 127, 129, 133, 138, 144, 153, 160, 162, 171, 174, 175, 186, 187, 195, 199, 202, 204, 220, 222, 223, 231, 243, 246, 252

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 30 2009

nonn

			a(1)=2 since 2^3-2^2-1=3 is prime.
a(2)=3 since 3^3-3^2-1=17 is prime.
a(3)=4 since 4^3-4^2-1=47 is prime.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A087908, A162291 (corresponding primes), A111501.

lst={};Do[s=n^3-n^2;If[PrimeQ[s-1],AppendTo[lst,n]],{n,6!}];lst

a(n)^2 * ( a(n)-1 )-1 = A162291(n).

Comments moved to the examples by R. J. Mathar, Sep 11 2009

A162294 Numbers k such that k^3-k^2-k-1 is prime.

Original entry on oeis.org

4, 6, 8, 12, 16, 22, 28, 34, 44, 50, 54, 56, 58, 76, 78, 88, 110, 112, 118, 134, 138, 156, 162, 166, 168, 170, 188, 190, 200, 204, 208, 210, 226, 230, 236, 244, 250, 268, 274, 302, 310, 314, 322, 324, 340, 344, 356, 364, 368, 378, 382, 390, 398, 400, 420, 424

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 30 2009

nonn

			k=4 is in the sequence because 4^3-4^2-4-1=43 is prime.
k=6 is in the sequence because 6^3-6^2-6-1=173 is prime.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A087908, A111501, A162291, A162293, A162295 (corresponding primes).

lst={};Do[p=n^3-n^2-n-1;If[PrimeQ[p],AppendTo[lst,n]],{n,2,6!}];lst

k^3-k^2-k-1 = A162295(n), where k=a(n).

Sum_{i=1..n} a(i) = Sum_{i=1..n} i * ( pi(i^3 - i^2 - i - 1) - pi(i^3 - i^2 - i - 2) ). - Wesley Ivan Hurt, May 24 2013

Edited by R. J. Mathar, Jul 02 2009

A162295 Primes of the form k^3-k^2-k-1.

Original entry on oeis.org

43, 173, 439, 1571, 3823, 10141, 21139, 38113, 83203, 122449, 154493, 172423, 191689, 433123, 468389, 673639, 1318789, 1392271, 1628989, 2388013, 2608889, 3771923, 4225121, 4546573, 4713239, 4883929, 6609139, 6822709, 7959799

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 30 2009

nonn

			a(1)=4^3-4^2-4-1=43. a(2)=6^3-6^2-6-1=173.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A087908, A162291, A111501, A162293, A162294, A162294 (corresponding k).

lst={};Do[p=n^3-n^2-n-1;If[PrimeQ[p],AppendTo[lst,p]],{n,2,6!}];lst

a(n)=k^3-k^2-k-1 where k=A162294(n).

Edited by R. J. Mathar, Jul 02 2009

A162292 Primes of the form k^3-k^2+1, k>0.

Original entry on oeis.org

5, 19, 101, 181, 449, 2029, 2549, 8821, 13249, 16901, 21169, 23549, 34849, 38149, 41651, 45361, 62401, 77659, 89101, 108289, 115249, 122501, 130051, 163351, 191749, 433201, 505601, 564899, 697049, 720901, 795709, 875521, 960499, 990001

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 30 2009

nonn

			a(1)=2^3-2^2+1=5. a(2)=3^3-3^2+1=19. a(3)=5^3-5^2+1=101.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A087908, A162291, A111501 (corresponding k).

lst={};Do[s=n^3-n^2;If[PrimeQ[s+1],AppendTo[lst,s+1]],{n,4*5!}];lst

a(n)= A111501(n)^3-A111501(n)^2+1 .

Edited by R. J. Mathar, Jul 02 2009

A239115 Numbers n such that (n-1)*n^2-1 and n^2-(n-1) are both prime.

Original entry on oeis.org

2, 3, 4, 6, 7, 9, 13, 18, 21, 22, 58, 67, 79, 90, 100, 106, 111, 118, 120, 144, 162, 174, 195, 204, 246, 273, 279, 345, 393, 403, 406, 435, 436, 526, 541, 567, 613, 625, 636, 702, 721, 729, 736, 744, 762, 763, 865, 898, 961, 970, 993, 1059, 1099, 1117, 1131

Offset: 1

Ilya Lopatin following a suggestion from Juri-Stepan Gerasimov, Mar 10 2014,

nonn

Numbers n such that (n^3-n^2-1)*(n^2-n+1) is semiprime.
Intersection of A162293 and A055494.
Primes in this sequence: 2, 3, 7, 13, 67, 79, 541, 613, 1117, ...
Squares in this sequence: 4, 9, 100, 144, 961, ...

			13 is in this sequence because (13-1)*13^2-1 = 2027 and 13^2-(13-1) = 157 are both prime.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A162291, A002383, A239135, A239326.

k:=1;
    for n in [1..1000] do
     if IsPrime(k*(n-1)*n^2-1) and IsPrime(k*n^2-n+1) then
            n;
      end if;
    end for; // Juri-Stepan Gerasimov, Mar 18 2014

Select[Range[1000], PrimeQ[#^3 - #^2 - 1] && PrimeQ[#^2 - # + 1] &] (* Giovanni Resta, Mar 10 2014 *)
Select[Range[1200],PrimeOmega[#^5-2#^4+2#^3-2#^2+#-1]==2&] (* Harvey P. Dale, Sep 24 2014 *)

isok(n) = isprime(n^3-n^2-1) && isprime(n^2-n+1); \\ Michel Marcus, Mar 10 2014

More terms from Giovanni Resta, Mar 10 2014

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A162293 Numbers k such that k^2*(k-1)-1 is prime.

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A162294 Numbers k such that k^3-k^2-k-1 is prime.

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A162295 Primes of the form k^3-k^2-k-1.

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A162292 Primes of the form k^3-k^2+1, k>0.

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A239115 Numbers n such that (n-1)*n^2-1 and n^2-(n-1) are both prime.

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