A182253 -id:A182253 - cp's OEIS frontend

A053183 Primes of the form p^2 + p + 1 when p is prime.

7, 13, 31, 307, 1723, 3541, 5113, 8011, 10303, 17293, 28057, 30103, 86143, 147073, 459007, 492103, 552793, 579883, 598303, 684757, 704761, 735307, 830833, 1191373, 1204507, 1353733, 1395943, 1424443, 1482307, 1886503, 2037757

Offset: 1

Enoch Haga, Mar 01 2000

Also primes in A001001. - Philippe Deléham, Feb 21 2004
This sequence is a subsequence of A002383. These numbers are repunit primes 111_n, so they are Brazilian primes belonging to A085104. - Bernard Schott, Dec 21 2012
Also, primes in A060800. - Zak Seidov, Mar 21 2014
Also subsequence of A002061, A193574. - Hartmut F. W. Hoft, May 05 2017
As p^2 + p + 1 is the sum of divisors of p^2 for any prime p, this is a subsequence of A023195. - Peter Munn, Feb 16 2018

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A002383, A002384, A023195, A053182, A060800, A065764, A085104, A182253, A185632.

Cf. A002061, A193574. - Hartmut F. W. Hoft, May 05 2017

a053183[n_] := Select[Map[Prime[#]^2 + Prime[#] + 1&, Range[n]], PrimeQ]
a053183[225] (* data *) (* Hartmut F. W. Hoft, May 05 2017 *)
Select[Table[p^2+p+1,{p,Prime[Range[300]]}],PrimeQ] (* Harvey P. Dale, Aug 15 2017 *)

a(n) = A053182(n)^2 + A053182(n) + 1.

A185632 Primes of the form n^2 + n + 1 where n is nonprime.

Original entry on oeis.org

3, 43, 73, 157, 211, 241, 421, 463, 601, 757, 1123, 1483, 2551, 2971, 3307, 3907, 4423, 4831, 5701, 6007, 6163, 6481, 8191, 9901, 11131, 12211, 12433, 13807, 14281, 19183, 20023, 20593, 21757, 22651, 23563, 24181, 26083, 26407, 27061, 28393, 31153, 35533

Offset: 1

Bernard Schott, Dec 18 2012

nonn

These are the primes associated with A182253.
All these numbers are in A002383 but not in A053183.
All the numbers n^2 + n + 1 = 111_n with n >= 2 are by definition Brazilian numbers: A125134. See Links: "Les nombres brésiliens" - Section V.5 page 35.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38.
Bernard Schott, Les nombres brésiliens, Reprinted from Quadrature, no. 76, avril-juin 2010, pages 30-38, included here with permission from the editors of Quadrature.

Cf. A002383, A002384, A053182, A053183, A182253.

Cf. A085104, A125134.

Select[Table[If[PrimeQ[n],Nothing,n^2+n+1],{n,200}],PrimeQ] (* Harvey P. Dale, Apr 02 2023 *)

lista(nn) = {for (n = 1, nn, if (! isprime(n) && isprime(p = n^2 + n + 1), print1(p, ", ");););} \\ Michel Marcus, Sep 04 2013

A192321 Semiprime numbers n such that n^2 + n + 1 is prime.

Original entry on oeis.org

6, 14, 15, 21, 33, 38, 57, 62, 69, 77, 111, 119, 141, 143, 155, 161, 194, 203, 206, 209, 215, 218, 278, 287, 309, 314, 329, 381, 395, 398, 447, 453, 489, 533, 537, 551, 554, 566, 579, 626, 635, 671, 755, 785, 818, 878, 899, 959, 974, 993, 1007, 1011, 1041, 1067, 1077, 1133, 1142, 1149, 1191, 1202, 1263

Offset: 1

Jonathan Vos Post, Dec 19 2012

			309 is in the sequence because 309 is semiprime (309 = 3 * 103) and 309^2 + 309 + 1 = 95791 is prime.

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

Intersection of A182253 and A001358.

Subsequence of A182253.

Select[Range[2000],PrimeOmega[#]==2&&PrimeQ[#^2+#+1]&] (* Harvey P. Dale, Feb 26 2013 *)

issemi(n)=bigomega(n)==2
is(n)=isprime(n^2+n+1) && issemi(n) \\ Charles R Greathouse IV, Jun 13 2017

A308238 Nonprimes k such that k^10 + k^9 + k^8 + k^7 + k^6 + k^5 + k^4 + k^3 + k^2 + k + 1 is prime.

Original entry on oeis.org

1, 20, 21, 30, 60, 86, 172, 195, 212, 224, 258, 268, 272, 319, 339, 355, 365, 366, 390, 398, 414, 480, 504, 534, 539, 543, 567, 592, 626, 654, 735, 756, 766, 770, 778, 806, 812, 874, 943, 973, 1003, 1036, 1040, 1065, 1194, 1210, 1239, 1243, 1264, 1309, 1311

Offset: 1

Bernard Schott, May 16 2019

nonn

A240693 Union {this sequence} = A162862.

The corresponding prime numbers, (11111111111)_k, are Brazilian primes and belong to A085104 and A285017 (except 11).

			(11111111111)_20 = (20^11 - 1)/19 = 10778947368421 is prime, thus 20 is a term.

Cf. A162861, A162862, A240693, A286301.
Intersection of A064108 and A285017.
Similar to A182253 for k^2+k+1, A286094 for k^4+k^3+k^2+k+1, A288939 for k^6+k^5+k^4+k^3+k^2+k+1.

[1] cat [n:n in [2..1500]|not IsPrime(n) and IsPrime(Floor((n^11-1)/(n-1)))]; // Marius A. Burtea, May 16 2019

filter:= n -> not isprime(n) and isprime((n^11-1)/(n-1)) : select(filter, [$2..5000]);

Select[Range@ 1320, And[! PrimeQ@ #, PrimeQ@ Total[#^Range[0, 10]]] &] (* Michael De Vlieger, Jun 09 2019 *)

isok(n) = !isprime(n) && isprime(polcyclo(11, n)); \\ Michel Marcus, May 19 2019

cp's OEIS Frontend

A053183 Primes of the form p^2 + p + 1 when p is prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A185632 Primes of the form n^2 + n + 1 where n is nonprime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A192321 Semiprime numbers n such that n^2 + n + 1 is prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A308238 Nonprimes k such that k^10 + k^9 + k^8 + k^7 + k^6 + k^5 + k^4 + k^3 + k^2 + k + 1 is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Maple

Mathematica

PARI