A193574 -id:A193574 - 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.

A190527 Primes of the form p^4 + p^3 + p^2 + p + 1, where p is prime.

Original entry on oeis.org

31, 2801, 30941, 88741, 292561, 732541, 3500201, 28792661, 39449441, 48037081, 262209281, 1394714501, 2666986681, 3276517921, 4802611441, 5908670381, 12936304421, 16656709681, 19408913261, 24903325661, 37226181521, 43713558101, 52753304641, 64141071121, 96427561501, 100648118041

Offset: 1

Bernard Schott, Dec 20 2012

nonn

These primes are generated by exactly A065509, cf. 2nd formula.
These numbers are repunit primes 11111_p, so they are Brazilian primes (A085104).
When p^4 + p^3 + p^2 + p + 1 = sigma(p^4) is prime, then it equals A193574(p^4), so that this sequence is a subsequence of A193574; by definition it is also a subsequence of A053699 and A131992. - Hartmut F. W. Hoft, May 05 2017

			a(3) = 30941 = 11111_13 = 13^4 + 13^3 + 13^2 + 13^1 + 1 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1100

Cf. A049409 (n^4 + ... + 1 is prime), A065509 (primes among these n), A193574.
Subsequence of A088548 (primes n^4 + ... + 1) and A085104 ("Brazilian" primes, of the form 1 + n + n^2 + ... + n^k).
Intersection of A000040 (primes) and A131992 (p^4 + ... + 1), subsequence of A053699 (n^4 + ... + 1).

[p: p in PrimesUpTo(600) | IsPrime(p) where p is p^4 +p^3+p^2+p+1]; // Vincenzo Librandi, May 06 2017

a190527[n_] := Select[Map[(Prime[#]^5-1)/(Prime[#]-1)&, Range[n]], PrimeQ]
a190527[100] (* data *) (* Hartmut F. W. Hoft, May 05 2017 *)
Select[#^4 + #^3 + #^2 + # + 1 &/@Prime[Range[100]], PrimeQ] (* Vincenzo Librandi, May 06 2017 *)

[q|p<-primes(100),ispseudoprime(q=(p^5-1)\(p-1))]
A190527_vec(N)=[(p^5-1)\(p-1)|p<-A065509_vec(N)] \\ M. F. Hasler, Mar 03 2020

a(n) = A193574(A065509(n)^4). - Hartmut F. W. Hoft, May 08 2017

a(n) = A053699(A065509(n)) = A000203(A065509(n)^4). - M. F. Hasler, Mar 03 2020

a(7) corrected and a(18)-a(26) added by Hartmut F. W. Hoft, May 05 2017

Edited by M. F. Hasler, Mar 06 2020

A194257 Primes of the form p^6 + p^5 + p^4 + p^3 + p^2 + p + 1 when p is prime.

Original entry on oeis.org

127, 1093, 19531, 5229043, 25646167, 917087137, 52379047267, 153436090543, 502628805631, 11016462577051, 18871143464293, 251059142817757, 1812169199976451, 1940350890330343

Offset: 1

Bernard Schott, Dec 21 2012

nonn

These primes are generated by exactly A163268.
This sequence is included in A088550.
These numbers are repunit primes 1111111_n, so they are Brazilian primes and are terms of A085104.
Subsequence of A088550. - Hartmut F. W. Hoft, May 05 2017

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A085104, A088550, A100330, A163268.

select(isprime, map(p -> add(p^i,i=0..6), select(isprime, [2,seq(i,i=3..1000,2)]))); # Robert Israel, May 05 2017

a194257[n_] := Select[Map[(Prime[#]^7-1)/(Prime[#]-1)&, Range[n]], PrimeQ]
a194257[70] (* data *) (* Hartmut F. W. Hoft, May 05 2017 *)
Select[Table[Total[p^Range[0,6]],{p,Prime[Range[100]]}],PrimeQ] (* Harvey P. Dale, Mar 09 2024 *)

a(n) = A193574(A163268(n)^6). - Hartmut F. W. Hoft, May 08 2017

A286301 Primes of the form p^10 + p^9 + p^8 + p^7 + p^6 + p^5 + p^4 + p^3 + p^2 + p + 1 when p is prime.

Original entry on oeis.org

12207031, 2141993519227, 178250690949465223, 2346320474383711003267, 398341412240537151131351, 79545183674814239059370551, 494424256962371823779424877, 8271964541879648991904246901, 32142180034067960734115528951, 91264002187709396686868598317

Offset: 1

Hartmut F. W. Hoft, May 05 2017

nonn

			Prime number 12207031 = Sum_{i=0..10} 5^i is the first in the sequence since 23 divides 88573 = Sum_{i=0..10} 3^i as well as 2047 = Sum_{i=0..10} 2^i.

Subsequence of A060885, A162861 and A193574.

Cf. A162862, A198244, A240693.

a286301[n_] := Select[Map[(Prime[#]^11-1)/(Prime[#]-1)&, Range[n]], PrimeQ]
a286301[150] (* data *)

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

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A190527 Primes of the form p^4 + p^3 + p^2 + p + 1, where p is prime.

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A194257 Primes of the form p^6 + p^5 + p^4 + p^3 + p^2 + p + 1 when p is prime.

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A286301 Primes of the form p^10 + p^9 + p^8 + p^7 + p^6 + p^5 + p^4 + p^3 + p^2 + p + 1 when p is prime.

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