A137530 -id:A137530 - cp's OEIS frontend

A212656 a(n) = 5*n^2 + 1.

1, 6, 21, 46, 81, 126, 181, 246, 321, 406, 501, 606, 721, 846, 981, 1126, 1281, 1446, 1621, 1806, 2001, 2206, 2421, 2646, 2881, 3126, 3381, 3646, 3921, 4206, 4501, 4806, 5121, 5446, 5781, 6126, 6481, 6846, 7221, 7606, 8001, 8406, 8821, 9246, 9681, 10126, 10581, 11046, 11521, 12006, 12501

Offset: 0

Alonso del Arte, May 23 2012

Z[sqrt(-5)] is not a unique factorization domain, and some of the numbers in this sequence have two different factorizations in that domain, e.g., 21 = 3 * 7 = (1 + 2*sqrt(-5))*(1 - 2*sqrt(-5)). And of course some primes in Z are composite in Z[sqrt(-5)], like 181 = (1 + 6*sqrt(-5))*(1 - 6*sqrt(-5)).

These are pentagonal-star numbers. - Mario Cortés, Oct 26 2020

Benjamin Fine & Gerhard Rosenberger, Number Theory: An Introduction via the Distribution of Primes, Boston: Birkhäuser, 2007, page 268.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A033429, A134538, A002522, A053755, A056107, A058331.

Cf. A137530 (primes of the form 1+5*n^2).

[5*n^2 + 1: n in [0..50]]; // Vincenzo Librandi, Jul 10 2012

Table[5n^2 + 1, {n, 0, 49}]
LinearRecurrence[{3,-3,1},{1,6,21},60] (* Harvey P. Dale, Apr 04 2017 *)

a(n)=5*n^2+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 5*n^2 + 1 = (1 + n*sqrt(-5))*(1 - n*sqrt(-5)).
G.f.: (1+3*x+6*x^2)/(1-x)^3. - Bruno Berselli, May 23 2012
a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - Vincenzo Librandi, Jul 10 2012
From Amiram Eldar, Jul 15 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(5))*coth(Pi/sqrt(5)))/2.
Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(5))*csch(Pi/sqrt(5)))/2. (End)
a(n) = A005891(n-1) + 5*A000217(n). - Mario Cortés, Oct 26 2020
From Amiram Eldar, Feb 05 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(5))*sinh(sqrt(2/5)*Pi).
Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(5))*csch(Pi/sqrt(5)).(End)
E.g.f.: exp(x)*(1 + 5*x + 5*x^2). - Stefano Spezia, Feb 05 2021

A138218 Numbers k such that 180k^2 + 1 is prime.

Original entry on oeis.org

1, 3, 6, 7, 14, 17, 18, 22, 24, 25, 27, 28, 29, 31, 32, 41, 46, 48, 50, 52, 55, 59, 62, 64, 66, 67, 76, 77, 83, 85, 87, 88, 92, 94, 95, 97, 102, 106, 108, 118, 123, 134, 136, 139, 140, 141, 147, 148, 154, 155, 157, 162, 165, 167, 179, 181, 192, 193, 199, 202, 203, 207

Offset: 1

Zak Seidov, May 05 2008

Or, numbers j arising in A137530.

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

Cf. A137530.

[n: n in [0..10000]|IsPrime(1+180*n^2)] // Vincenzo Librandi, Dec 13 2010

Select[Range[250],PrimeQ[180#^2+1]&] (* Harvey P. Dale, May 26 2024 *)

is(n)=isprime(1+180*n^2) \\ Charles R Greathouse IV, Jun 13 2017

a(n) = sqrt((A137530(n) - 1)/180).

A207837 Primes of the form 5*k^4 + 1.

Original entry on oeis.org

6481, 103681, 844480081, 1036800001, 55099802881, 63727534081, 115672050001, 155584800001, 307529920081, 322620641281, 425152800001, 1019640545281, 1633266996481, 1739461754881, 2489356800001, 2634683086081, 2944329626881, 5285935072081, 6360160441681

Offset: 1

Bruno Berselli, Feb 21 2012

Also primes of the form 6480*k^4+1.

Vincenzo Librandi, Table of n, a(n) for n = 1..1500

Cf. A207838 (values of k).
Subsequence of A137530.
Primes of the form k*n^(k-1)+1: A065091, A002648, A199307.

[6480*n^4+1: n in [1..181] | IsPrime(6480*n^4+1)];

Select[5 Range[1086]^4 + 1, PrimeQ] (* by definition *)

for(n=1, 181, r=6480*n^4+1; if(isprime(r), print1(r", ")));

a(n) = 5*A207838(n)^4 + 1. - Paul F. Marrero Romero, Dec 07 2023

A212707 Semiprimes of the form 5*n^2 + 1.

Original entry on oeis.org

6, 21, 46, 321, 501, 721, 1126, 2206, 2881, 3646, 3921, 4501, 7606, 10581, 11521, 13521, 14581, 15681, 16246, 18001, 19846, 20481, 21781, 23806, 24501, 27381, 30421, 32001, 38721, 40501, 42321, 48021, 61606, 64981, 72001, 79381, 83206, 89781, 106581, 121681

Offset: 1

Jonathan Vos Post, May 24 2012

This is to A137530 (primes of form 1+5n^2) as semiprimes A001358 are to primes A000040. Since Z[sqrt(-5)] is not a unique factorization domain, some numbers of form 1+5n^2 are primes in Z but composite in Z[sqrt(-5)]; some values in this sequence are semiprimes in Z but have a different number than 2 of prime factors in Z[sqrt(-5)].

			a(6) = 721 = 1 + 5*(12^2) = 7 * 103.

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

Cf. A001222, A001358, A137530, A212656 (5*n^2 + 1).

IsSemiprime:= func; [s: n in [1..180] | IsSemiprime(s) where s is 5*n^2 + 1]; // Vincenzo Librandi, Sep 22 2012

SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])]; Select[Table[5*n^2 + 1, {n, 200}], SemiPrimeQ] (* T. D. Noe, May 24 2012 *)
Select[Table[5*n^2 + 1, {n, 180}], PrimeOmega[#] == 2&] (* Vincenzo Librandi, Sep 22 2012 *)

A212656 INTERSECTION A001358.

{k such that 5*n^2 + 1 for a natural number n, and bigomega(k) = A001222(k) = 2}.

Extended by T. D. Noe, May 24 2012

A247965 a(n) is the smallest number k such that m*k^2+1 is prime for all m = 1 to n.

Original entry on oeis.org

1, 1, 6, 3240, 113730, 30473520, 3776600100, 16341921960, 3332396388090

Offset: 1

Michel Lagneau, Sep 28 2014

Conjecture : the sequence is infinite.

a(10) > 15466500000000. a(11) > 107669100000000. - Hiroaki Yamanouchi, Oct 01 2014

			a(3)=6 because 6^2+1 = 37, 2*6^2+1 = 73 and 3*6^2+1 = 109 are prime numbers.
The resulting primes begin like this:
2;
2, 3;
37, 73, 109;
10497601, 20995201, 31492801, 41990401;
... - _Michel Marcus_, Sep 29 2014

Cf. A002496, A090698, A002648, A137530, A090687, A201602, A090685, A156226.

for n from 1 to 6 do:
  ii:=0:
   for k from 1 to 10^10 while(ii=0) do:
     ind:=0:
       for m from 1 to n do:
         p:=m*k^2+1:
          if type(p,prime) then
           ind:=ind+1:
           fi:
        od:
       if ind=n then
        ii:=1:printf ( "%d %d \n",n,k):
       fi:
    od:
  od:

a(n)=k=1;while(k,c=0;for(i=1,n,if(!ispseudoprime(i*k^2+1),c++;break));if(!c,return(k));if(c,k++))
n=1;while(n<10,print1(a(n),", ");n++) \\ Derek Orr, Sep 28 2014

a(7)-a(9) from Hiroaki Yamanouchi, Oct 01 2014

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A212656 a(n) = 5*n^2 + 1.

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A138218 Numbers k such that 180k^2 + 1 is prime.

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A207837 Primes of the form 5*k^4 + 1.

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A212707 Semiprimes of the form 5*n^2 + 1.

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A247965 a(n) is the smallest number k such that m*k^2+1 is prime for all m = 1 to n.

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