A224489 -id:A224489 - cp's OEIS frontend

A224490 Smallest k such that k2p(n)^2-1=q is prime and k2q^2-1 is also prime.

1, 1, 25, 9, 21, 3, 1, 16, 25, 136, 10, 33, 90, 250, 10, 55, 1, 9, 36, 75, 1, 4, 33, 406, 103, 15, 121, 4, 244, 78, 28, 19, 49, 105, 45, 34, 10, 46, 33, 4, 111, 15, 9, 36, 118, 66, 10, 13, 31, 76, 66, 36, 55, 15, 4, 48, 6, 66, 13, 34, 54, 64, 153, 1, 60, 48

Offset: 1

Pierre CAMI, Apr 08 2013

nonn

			1*2*2^2-1=7 prime q 1*2*7^2-1=97 also prime so a(1)=1.

Pierre CAMI, Table of n, a(n) for n = 1..10000

Cf. A224489, A224491, A224492.

a[n_] := For[k = 1, True, k++, p = Prime[n]; If[PrimeQ[q = k*2*p^2 - 1] && PrimeQ[k*2*q^2 - 1], Return[k]]]; Table[a[n], {n, 1, 66}] (* Jean-François Alcover, Apr 12 2013 *)

Typo in name fixed by Zak Seidov, Apr 11 2013

A224492 Smallest k such that k2p(n)^2-1=q is prime, k2q^2-1=r, k2r^2-1=s, k2r^2-1=t, r, s, and t are also prime.

Original entry on oeis.org

5103, 36189, 7315, 29608, 128115, 3496, 64590, 143079, 83919, 5586, 13209, 2833, 235339, 61621, 164349, 2668, 84574, 1140, 47335, 108079, 7978, 181366, 146140, 2616, 165864, 86100, 11455, 8925, 23191, 197938, 28194, 229309, 196236, 274186

Offset: 1

Pierre CAMI, Apr 08 2013

nonn

conjecture: a(n) exist for all n
t=k*2*(k*2*(k*2*(k*2*p(n)^2-1)^2-1)^2-1)^2-1
s=k*2*(k*2*(k*2*p(n)^2-1)^2-1)^2-1
r=k*2*(k*2*p(n)^2-1)^2-1
q=k*2*p(n)^2-1

Pierre CAMI, Table of n, a(n) for n = 1..80

Cf. A224489, A224490, A224491.

a[n_] := For[k = 1, True, k++, p = Prime[n]; If[PrimeQ[q = k*2*p^2 - 1] && PrimeQ[r = k*2*q^2 - 1] && PrimeQ[s = k*2*r^2 - 1] && PrimeQ[k*2*s^2 - 1], Return[k]]]; Table[Print[an = a[n]]; an, {n, 1, 34}] (* Jean-François Alcover, Apr 12 2013 *)

Typo in name fixed by Zak Seidov, Apr 11 2013

A224491 Smallest k such that k2p(n)^2-1=q is prime k2q^2-1=r k2r^2-1=s, r and s are also prime.

Original entry on oeis.org

705, 1, 306, 390, 2539, 526, 1939, 439, 7048, 286, 561, 985, 90, 2385, 2089, 328, 2266, 664, 4245, 2451, 453, 391, 411, 406, 4068, 4975, 8151, 199, 834, 4423, 169, 76, 5710, 861, 3930, 1659, 1246, 2838, 750, 153, 8664, 3730, 1195, 7815, 1746, 1735, 594, 985

Offset: 1

Pierre CAMI, Apr 08 2013

nonn

Pierre CAMI, Table of n, a(n) for n = 1..5600

Cf. A224489, A224490, A224492.

a[n_] := For[k = 1, True, k++, p = Prime[n]; If[PrimeQ[q = k*2*p^2 - 1] && PrimeQ[r = k*2*q^2 - 1] && PrimeQ[k*2*r^2 - 1], Return[k]]]; Table[a[n], {n, 1, 48}] (* Jean-François Alcover, Apr 12 2013 *)

Typo in name fixed by Zak Seidov, Apr 11 2013

A224609 Smallest j such that 2jprime(n)^3-1 is prime.

Original entry on oeis.org

2, 1, 2, 7, 2, 7, 8, 6, 8, 5, 1, 3, 11, 1, 9, 3, 5, 1, 3, 15, 7, 3, 8, 8, 12, 2, 15, 3, 10, 2, 3, 12, 12, 1, 6, 6, 9, 3, 5, 2, 5, 1, 5, 10, 57, 1, 21, 1, 15, 9, 2, 3, 1, 5, 5, 3, 15, 6, 7, 5, 25, 6, 12, 11, 6, 5, 1, 9, 2, 19, 5, 9, 27, 1, 3, 11, 3, 15, 2, 6, 21

Offset: 1

Pierre CAMI, Apr 12 2013

nonn

We are searching smallest j such that j*prime(n)*2*p(n)^2-1 is prime, for A224489 it is smallest k such that k*2*prime(n)^2-1 is prime, so here we replace smallest k by smallest j*prime(n).

			1*2*2^3-1= 15 is composite; 2*2*2^3-1= 31 is prime, so a(1)=2 as p(1)=2.
1*2*3^3-1=53 is prime, so a(2)=1 as p(2)=3.
1*2*5^3-1=249 is composite; 2*2*5^3=499 is prime, so a(3)=2 as p(3)=5.

Pierre CAMI, Table of n, a(n) for n = 1..10000

Cf. A224489.

S:=[];
j:=1;
for n in [1..100] do
  while not IsPrime(2*j*NthPrime(n)^3-1) do
       j:=j+1;
  end while;
  Append(~S, j);
  j:=1;
end for;
S; // Bruno Berselli, Apr 18 2013

jmax = 10^5 (* sufficient up to 10^5 terms *); a[n_] := For[j = 1, j <= jmax, j++, p = Prime[n]; If[PrimeQ[j*2*p^3 - 1], Return[j]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Apr 18 2013 *)

a(n)=my(P=2*prime(n)^3,j);while(!isprime(j++*P-1),);j \\ Charles R Greathouse IV, Apr 18 2013

a(n) = A053989(2p^3) where p is the n-th prime. - Charles R Greathouse IV, Apr 18 2013

A224494 Smallest k such that k2p(n)^2+1=q is prime and k2q^2+1 is also prime.

Original entry on oeis.org

5, 2, 29, 41, 9, 2, 71, 30, 32, 6, 35, 11, 6, 50, 2, 20, 9, 120, 56, 21, 9, 75, 90, 51, 51, 29, 107, 9, 74, 155, 116, 11, 29, 86, 116, 35, 200, 12, 11, 39, 9, 105, 51, 422, 36, 65, 6, 32, 27, 44, 9, 41, 14, 116, 266, 41, 29, 5, 50, 95, 27, 71, 69, 330, 21, 194

Offset: 1

Pierre CAMI, Apr 08 2013

nonn

Pierre CAMI, Table of n, a(n) for n = 1..10000

Cf. A224489, A224490, A224491, A224492, A224493, A224495, A224496.

a[n_] := For[k = 1, True, k++, p = Prime[n]; If[PrimeQ[q = k*2*p^2 + 1] && PrimeQ[k*2*q^2 + 1], Return[k]]]; Table[ a[n] , {n, 1, 66}] (* Jean-François Alcover, Apr 12 2013 *)

A224495 Smallest k such that k2p(n)^2+1=q is prime 2kq^2+1=r 2kr^2+1=s, r and s are also prime.

Original entry on oeis.org

9, 126, 29, 237, 420, 2, 186, 30, 2349, 896, 1266, 147, 741, 140, 3021, 924, 19571, 896, 791, 11495, 32, 7016, 3522, 5336, 932, 5480, 107, 1439, 1770, 209, 4239, 1716, 477, 1196, 1446, 900, 9176, 1920, 2375, 39, 2351, 590, 2724, 422, 3171, 179, 1751, 426, 65

Offset: 1

Pierre CAMI, Apr 08 2013

nonn

Pierre CAMI, Table of n, a(n) for n = 1..5600

Cf. A224489, A224490, A224492, A224492, A224493, A224494, A224496.

a[n_] := For[k = 1, True, k++, p = Prime[n]; If[PrimeQ[q = k*2*p^2 + 1] && PrimeQ[r = k*2*q^2 + 1] && PrimeQ[k*2*r^2 + 1], Return[k]]]; Table[ a[n] , {n, 1, 49}] (* Jean-François Alcover, Apr 12 2013 *)

A224496 Smallest k such that k2p(n)^2+1=q is prime, k2q^2+1=r, k2r^2+1=s, k2r^2+1=t, r, s, and t are also prime.

Original entry on oeis.org

386, 2769, 96656, 5366, 420, 34454, 65039, 192215, 458367, 24735, 27155, 777, 736254, 80297, 279927, 113429, 650474, 238919, 8229, 1284345, 642789, 333141, 11510, 1009271, 932, 395126, 1202174, 25811, 204534, 16286, 22094, 2661131, 22530, 128225, 56225, 900

Offset: 1

Pierre CAMI, Apr 08 2013

nonn

Conjecture: a(n) exist for all n
t=k*2*(k*2*(k*2*(k*2*p(n)^2+1)^2+1)^2+1)^2+1
s=k*2*(k*2*(k*2*p(n)^2+1)^2+1)^2+1
r=k*2*(k*2*p(n)^2+1)^2+1
q=k*2*p(n)^2+1

Pierre CAMI, Table of n, a(n) for n = 1..80

Cf. A224489, A224490, A224491, A224492, A224193, A224494, A224495.

a[n_] := For[k = 1, True, k++, p = Prime[n]; If[PrimeQ[q = k*2*p^2 + 1] && PrimeQ[r = k*2*q^2 + 1] && PrimeQ[s = k*2*r^2 + 1] && PrimeQ[k*2*s^2 + 1], Return[k]]]; Table[ Print[an = a[n]]; an , {n, 1, 36}] (* Jean-François Alcover, Apr 12 2013 *)

A224493 Smallest k such that k2p(n)^2+1 is prime.

Original entry on oeis.org

2, 1, 2, 2, 3, 2, 6, 15, 12, 6, 8, 2, 5, 6, 2, 14, 3, 23, 2, 5, 2, 3, 5, 3, 6, 11, 2, 9, 3, 5, 6, 3, 14, 8, 5, 6, 2, 2, 5, 9, 8, 11, 3, 2, 11, 3, 6, 5, 6, 5, 2, 5, 3, 8, 15, 14, 3, 5, 20, 5, 6, 14, 14, 8, 5, 2, 8, 2, 6, 18, 14, 3, 6, 9, 5, 12, 3, 9, 15, 18, 6, 6, 3

Offset: 1

Pierre CAMI, Apr 08 2013

nonn

Pierre CAMI, Table of n, a(n) for n = 1..10000

Cf. A224489, A224490, A224491, A224492, A224494, A224495, A224496.

a[n_] := For[k = 1, True, k++, p = Prime[n]; If[PrimeQ[k*2*p^2 + 1], Return[k]]]; Table[ a[n] , {n, 1, 83}] (* Jean-François Alcover, Apr 12 2013 *)
sk[n_]:=Module[{k=1},While[!PrimeQ[2*k*n^2+1],k++];k]; Table[sk[n],{n,Prime[ Range[ 90]]}] (* Harvey P. Dale, Sep 22 2019 *)

cp's OEIS Frontend

A224490 Smallest k such that k*2*p(n)^2-1=q is prime and k*2*q^2-1 is also prime.

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A224492 Smallest k such that k*2*p(n)^2-1=q is prime, k*2*q^2-1=r, k*2*r^2-1=s, k*2*r^2-1=t, r, s, and t are also prime.

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A224491 Smallest k such that k*2*p(n)^2-1=q is prime k*2*q^2-1=r k*2*r^2-1=s, r and s are also prime.

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A224609 Smallest j such that 2*j*prime(n)^3-1 is prime.

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A224494 Smallest k such that k*2*p(n)^2+1=q is prime and k*2*q^2+1 is also prime.

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A224495 Smallest k such that k*2*p(n)^2+1=q is prime 2*k*q^2+1=r 2*k*r^2+1=s, r and s are also prime.

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A224496 Smallest k such that k*2*p(n)^2+1=q is prime, k*2*q^2+1=r, k*2*r^2+1=s, k*2*r^2+1=t, r, s, and t are also prime.

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A224493 Smallest k such that k*2*p(n)^2+1 is prime.

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A224490 Smallest k such that k2p(n)^2-1=q is prime and k2q^2-1 is also prime.

A224492 Smallest k such that k2p(n)^2-1=q is prime, k2q^2-1=r, k2r^2-1=s, k2r^2-1=t, r, s, and t are also prime.

A224491 Smallest k such that k2p(n)^2-1=q is prime k2q^2-1=r k2r^2-1=s, r and s are also prime.

A224609 Smallest j such that 2jprime(n)^3-1 is prime.

A224494 Smallest k such that k2p(n)^2+1=q is prime and k2q^2+1 is also prime.

A224495 Smallest k such that k2p(n)^2+1=q is prime 2kq^2+1=r 2kr^2+1=s, r and s are also prime.

A224496 Smallest k such that k2p(n)^2+1=q is prime, k2q^2+1=r, k2r^2+1=s, k2r^2+1=t, r, s, and t are also prime.

A224493 Smallest k such that k2p(n)^2+1 is prime.