cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A224489 Smallest k such that k*2*p(n)^2-1 is prime.

Original entry on oeis.org

1, 1, 3, 1, 1, 1, 1, 4, 4, 6, 4, 6, 1, 1, 9, 10, 1, 6, 4, 7, 1, 4, 3, 4, 3, 10, 4, 4, 1, 1, 1, 10, 1, 7, 6, 12, 1, 9, 6, 3, 1, 1, 6, 3, 1, 1, 1, 3, 3, 4, 4, 21, 4, 1, 3, 1, 6, 4, 1, 10, 3, 1, 15, 1, 3, 4, 9, 13, 10, 9, 1, 4, 1, 3, 1, 3, 12, 9, 6, 1, 1, 22, 4, 1
Offset: 1

Views

Author

Pierre CAMI, Apr 08 2013

Keywords

Examples

			1*2*2^2-1=7 is prime, p(1)=2 so a(1)=1.
1*2*3^2-1=17 is prime, p(2)=3 so a(2)=1.
1*2*5^2-1=49 is composite; 2*2*5^2-1=99 is composite; 3*2*5^2-1=149 is prime, p(3)=5 so a(3)=3.

Links

Crossrefs

Cf. A224490, A224491, A224492.

Programs

  • Magma
    S:=[];
k:=1;
for n in [1..90] do
  while not IsPrime(k*2*NthPrime(n)^2-1) do
       k:=k+1;
  end while;
  Append(~S, k);
  k:=1;
end for;
S; // Bruno Berselli, Apr 18 2013
  • Mathematica
    a[n_] := For[k = 1, True, k++, If[ PrimeQ[k*2*Prime[n]^2 - 1], Return[k]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Apr 10 2013 *)

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

Original entry on oeis.org

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

Views

Author

Pierre CAMI, Apr 08 2013

Keywords

Examples

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

Links

Crossrefs

Cf. A224489, A224491, A224492.

Programs

  • Mathematica
    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 *)

Extensions

Typo in name fixed by Zak Seidov, Apr 11 2013

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.

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

Views

Author

Pierre CAMI, Apr 08 2013

Keywords

Links

Crossrefs

Cf. A224489, A224490, A224492.

Programs

  • Mathematica
    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 *)

Extensions

Typo in name fixed by Zak Seidov, Apr 11 2013

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

Views

Author

Pierre CAMI, Apr 08 2013

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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 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.

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

Views

Author

Pierre CAMI, Apr 08 2013

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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 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.

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

Views

Author

Pierre CAMI, Apr 08 2013

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 k*2*p(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

Views

Author

Pierre CAMI, Apr 08 2013

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

A224612 Let p = prime(n). Smallest j such that q = j*2*p^3-1, r = j*p*2*q^2-1, s = j*p*2*r^2-1, and j*p*2*s^2-1 are prime numbers.

Original entry on oeis.org

29952, 12063, 1463, 6102, 11661, 49552, 639179, 2099290, 291248, 393186, 545251, 321303, 436641, 278295, 746832, 237852, 56490, 165901, 152847, 619755, 777177, 3410085, 117513, 2015421, 497170, 14750, 161190, 347039, 251835, 57536, 222, 2083286, 384944, 1228474, 3909960, 344164, 332224, 207751, 14060
Offset: 1

Views

Author

Pierre CAMI, Apr 12 2013

Keywords

Comments

Conjecture: a(n) exists for all n.

Crossrefs

Cf. A224492, A224609, A224610, A224611.

Programs

  • Maple
    f:= proc(n) local j,p,q,r,s;
    p:= ithprime(n);
    for j from 1 do
      q:= j*2*p^3-1; if not isprime(q) then next fi;
      r:= j*p*2*q^2-1; if not isprime(r) then next fi;
      s:= j*p*2*r^2-1; if not isprime(s) then next fi;
      if isprime(j*p*2*s^2-1) then return j fi;
    od
end proc;
map(f, [$1..25]); # Robert Israel, May 15 2025
  • Mathematica
    a[n_] := For[j = 1, j < 10^7, j++, p = Prime[n]; If[PrimeQ[q = j*2*p^3 - 1] && PrimeQ[r = j*2*p*q^2 - 1] && PrimeQ[s = j*2*p*r^2 - 1] && PrimeQ[j*2*p*s^2 - 1], Return[j]]]; Table[Print[an = a[n]]; an, {n, 1, 24}] (* Jean-François Alcover, Apr 12 2013 *)

Extensions

More terms from Jean-François Alcover, Apr 12 2013
Name clarified and more terms from Robert Israel, May 15 2025
Showing 1-8 of 8 results.

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