A125780 -id:A125780 - cp's OEIS frontend

A125855 Numbers k such that k+1, k+3, k+7 and k+9 are all primes.

4, 10, 100, 190, 820, 1480, 1870, 2080, 3250, 3460, 5650, 9430, 13000, 15640, 15730, 16060, 18040, 18910, 19420, 21010, 22270, 25300, 31720, 34840, 43780, 51340, 55330, 62980, 67210, 69490, 72220, 77260, 79690, 81040, 82720, 88810, 97840

Offset: 1

Artur Jasinski, Dec 12 2006

nonn

It seems that, with the exception of 4, all terms are multiples of 10. - Emeric Deutsch, Dec 24 2006
In fact, all terms except 4 are congruent to 10 (mod 30). - Franklin T. Adams-Watters, Jun 05 2014
For n > 1: a(n) = 10*A007811(n-1). - Reinhard Zumkeller, Jul 18 2014 [Comment corrected by Jens Kruse Andersen, Jul 19 2014]

			For k = 10, the numbers 10 + 1 = 11, 10 + 3 = 13, 10 + 7 = 17, 10 + 9 = 19 are prime. - _Marius A. Burtea_, May 18 2019

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A007530, A057015, A125779, A125780.

Cf. A010051, A245304 (subsequence), A007811.

a125855 n = a125855_list !! (n-1)
a125855_list = map (pred . head) $ filter (all (== 1) . map a010051') $
   iterate (zipWith (+) [1, 1, 1, 1]) [1, 3, 7, 9]
-- Reinhard Zumkeller, Jul 18 2014

[n:n in [1..100000]| IsPrime(n+1) and IsPrime(n+3) and IsPrime(n+7) and IsPrime(n+9)]; // Marius A. Burtea, May 18 2019

a:=proc(n): if isprime(n+1)=true and isprime(n+3)=true and isprime(n+7)=true and isprime(n+9)=true then n else fi end: seq(a(n),n=1..500000); # Emeric Deutsch, Dec 24 2006

Do[If[(PrimeQ[x + 1]) && (PrimeQ[x + 3]) && (PrimeQ[x + 7]) && (PrimeQ[x + 9]), Print[x]], {x, 1, 10000}]
(* Second program *)
Select[Range[10^5], Times @@ Boole@ Map[PrimeQ, # + {1, 3, 7, 9}] == 1 &] (* Michael De Vlieger, Jun 12 2017 *)
Select[Range[100000],AllTrue[#+{1,3,7,9},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 02 2018 *)

is(n) = my(v=[1, 3, 7, 9]); for(t=1, #v, if(!ispseudoprime(n+v[t]), return(0))); 1 \\ Felix Fröhlich, May 18 2019

a(n) = A007530(n) - 1. - R. J. Mathar, Jun 14 2017

More terms from Emeric Deutsch, Dec 24 2006

A125779 Numbers n such that n^4 + 1, n^4 + 3, n^4 + 7 and n^4 + 9 are all prime.

Original entry on oeis.org

83270, 519370, 939220, 1844170, 2263910, 2293460, 2429260, 2595980, 3133640, 3216530, 3474200, 3559760, 4787050, 5306720, 5505940, 6238780, 6889430, 6932770, 7320160, 8286340, 8427880, 8744290, 8961590, 9863440, 10871530

Offset: 1

Artur Jasinski, Dec 09 2006

nonn

Schinzel proved in 1958 that the set of primes of kind n^(2^k) + 1, n^(2^k) + 3, n^(2^k) + 7 and n^(2^k) + 9 is infinite for each number k > 0.

Sierpinski, W. Elementary theory of numbers. Warszawa 1964 Monografie Matematyczne Vol. 42.

Robert Israel, Table of n, a(n) for n = 1..500

Cf. A057015, A125780.

R:= NULL: count:= 0:
for k from 0 while count < 30 do
  for i in [20,30,40,50] do
    n:= 70*k + i;
    if isprime(n^4+1) and isprime(n^4+3) and isprime(n^4+7) and isprime(n^4+9) then
      count:= count+1; R:= R, n;
    fi
od od:
R; # Robert Israel, Feb 11 2021

Select[Range[109*10^5],AllTrue[#^4+{1,3,7,9},PrimeQ]&] (* Harvey P. Dale, May 15 2022 *)

Corrected and extended by Donovan Johnson, Apr 22 2008

A125856 a(n) = least number k such that k^(2^n)+1, k^(2^n)+3, k^(2^n)+7 and k^(2^n)+9 are all prime.

Original entry on oeis.org

4, 2, 83270, 5241160, 57171410, 359829200

Offset: 0

Artur Jasinski, Dec 12 2006

nonn

In 1958, Schinzel showed that for each n>0, there are infinitely many primes among the numbers k^(2^n)+{1,3,7, or 9}.

Sierpinski, W. Elementary theory of numbers. Warszawa 1964 Monografie Matematyczne Vol. 42.

Cf. A125855, A057015, A125779, A125780.

a(n) = {k = 1; while(!isprime(k^(2^n)+1) || !isprime(k^(2^n)+3) || !isprime(k^(2^n)+7) || !isprime(k^(2^n)+9), k++); k;} \\ Michel Marcus, Nov 03 2013

Edited by Don Reble, Dec 16 2006

One more term from Farideh Firoozbakht, Jan 01 2007

cp's OEIS Frontend

A125855 Numbers k such that k+1, k+3, k+7 and k+9 are all primes.

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A125779 Numbers n such that n^4 + 1, n^4 + 3, n^4 + 7 and n^4 + 9 are all prime.

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Maple

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A125856 a(n) = least number k such that k^(2^n)+1, k^(2^n)+3, k^(2^n)+7 and k^(2^n)+9 are all prime.

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