A201602 -id:A201602 - cp's OEIS frontend

A247541 a(n) = 7*n^2 + 1.

1, 8, 29, 64, 113, 176, 253, 344, 449, 568, 701, 848, 1009, 1184, 1373, 1576, 1793, 2024, 2269, 2528, 2801, 3088, 3389, 3704, 4033, 4376, 4733, 5104, 5489, 5888, 6301, 6728, 7169, 7624, 8093, 8576, 9073, 9584, 10109, 10648, 11201, 11768, 12349, 12944, 13553

Offset: 0

Karl V. Keller, Jr., Sep 18 2014

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A201602 (primes of the form 7n^2 + 1).

[7*n^2+1: n in [0..50]]; // Vincenzo Librandi, Sep 19 2014

a247541[n_Integer] := 7 n^2 + 1; a247541 /@ Range[0, 120] (* Michael De Vlieger, Sep 18 2014 *)
CoefficientList[Series[(1 + 5 x + 8 x^2)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Sep 19 2014 *)
LinearRecurrence[{3,-3,1},{1,8,29},50] (* Harvey P. Dale, Jun 09 2015 *)

vector(100,n,7*(n-1)^2+1) \\ Derek Orr, Sep 18 2014

for n in range (0,500) : print (7*n**2+1)

G.f.: (1 + 5*x + 8*x^2)/(1 - x)^3. - Vincenzo Librandi, Sep 19 2014
From Amiram Eldar, Jul 15 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(7))*coth(Pi/sqrt(7)))/2.
Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(7))*csch(Pi/sqrt(7)))/2. (End)
From Amiram Eldar, Feb 05 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(7))*sinh(sqrt(2/7)*Pi).
Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(7))*csch(Pi/sqrt(7)). (End)
E.g.f.: exp(x)*(1 + 7*x + 7*x^2). - Stefano Spezia, Feb 05 2021

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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A247541 a(n) = 7*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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