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 1-6 of 6 results.

A110589 Primes p such that 2*q+3 = p^2, where q is prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 41, 47, 67, 73, 79, 97, 101, 103, 109, 151, 157, 197, 211, 227, 233, 239, 257, 263, 293, 307, 331, 337, 349, 353, 359, 367, 389, 397, 409, 443, 449, 463, 487, 491, 509, 521, 547, 569, 571, 587, 613, 619, 653, 661, 673, 727
Offset: 1

Views

Author

Walter Kehowski, Sep 13 2005

Keywords

Links

Crossrefs

Cf. A098828, A109358.

Programs

  • Magma
    [p: p in PrimesUpTo(1000) | IsPrime((p^2-3) div 2)]; // Vincenzo Librandi, Mar 19 2015
  • Maple
    ispower := proc(n,b) andmap(proc(w) evalb(w[2] mod b = 0) end, ifactors(n)[2]) end: a:=2: SQRTP||a:=[]; for z from 1 to 1 do for n from 1 to 10000 do p:=ithprime(n); m:=a*p+a+1; if ispower(m,2) and isprime(sqrt(m)) then SQRTP||a:=[op(SQRTP||a),sqrt(m)] fi od; od; SQRTP||a;
  • Mathematica
    fQ[n_] := PrimeQ[(n^2 - 3)/2]; Select[ Prime@ Range@129, fQ@# &] (* Robert G. Wilson v, Jun 19 2006 *)
Select[Table[Sqrt[2 Prime[n] + 3], {n, 1, 30000}], PrimeQ] (* Vincenzo Librandi, Mar 19 2015 *)

Extensions

More terms from Robert G. Wilson v, Jun 19 2006

A109358 Square root of squares of form 2*p + 3, where p is prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 19, 23, 25, 29, 31, 37, 41, 47, 55, 65, 67, 73, 77, 79, 91, 97, 101, 103, 109, 119, 143, 151, 155, 157, 161, 187, 197, 205, 209, 211, 221, 227, 233, 235, 239, 245, 253, 257, 263, 265, 275, 287, 289, 293, 299, 305, 307, 323, 331, 337, 349, 353
Offset: 1

Views

Author

Giovanni Teofilatto, Aug 22 2005

Keywords

Comments

Numbers n such that (n^2-3)/2 is prime. - Robert Israel, Jan 22 2018

Links

Crossrefs

Cf. A098828, A109367.

Programs

  • Maple
    select(t -> isprime((t^2-3)/2), [seq(n,n=1..1000,2)]); # Robert Israel, Jan 22 2018
  • Mathematica
    Select[Table[(2Prime[n] + 3)^(1/2), {n, 7000}], IntegerQ] (* Ray Chandler, Aug 25 2005 *)

Formula

a(n) = SQRT(A109367(n)).

Extensions

Extended by Ray Chandler, Aug 25 2005

A109367 Squares of the form 2*p + 3, where p is a prime.

Original entry on oeis.org

9, 25, 49, 121, 169, 361, 529, 625, 841, 961, 1369, 1681, 2209, 3025, 4225, 4489, 5329, 5929, 6241, 8281, 9409, 10201, 10609, 11881, 14161, 20449, 22801, 24025, 24649, 25921, 34969, 38809, 42025, 43681, 44521, 48841, 51529, 54289, 55225, 57121
Offset: 1

Views

Author

Giovanni Teofilatto, Aug 23 2005

Keywords

Comments

The first seven terms are primes squared: 3^2, 5^2, 7^2, 11^2, 13^2, 17^2, 19^2, 23^2.

Links

Crossrefs

Cf. A098828, A109358.

Programs

  • Mathematica
    Select[Table[(2Prime[n] + 3)^(1/2), {n, 3500}], IntegerQ]^2 (* Ray Chandler, Aug 25 2005 *)
Select[2*Prime[Range[3500]]+3,IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, Sep 07 2020 *)
  • PARI
    is(n)=issquare(n) && n%2 && isprime(n\2-1) \\ Charles R Greathouse IV, Aug 06 2013

Formula

a(n) = A109358(n)^2 = 2*A098828(n) + 3.

Extensions

Extended by Ray Chandler and Robert G. Wilson v, Aug 25 2005

A099007 Primes of the form 6n^2 - 2n - 1.

Original entry on oeis.org

3, 19, 47, 139, 367, 467, 839, 1319, 1699, 1907, 3407, 4003, 4987, 6079, 7703, 10499, 11527, 13159, 16747, 17387, 19379, 23687, 25219, 26003, 30103, 32707, 33599, 35419, 38239, 44203, 50599, 53959, 55103, 57427, 62219, 69767, 72379, 76387
Offset: 1

Views

Author

Giovanni Teofilatto, Nov 07 2004

Keywords

Comments

All terms are == 3 (mod 4).

Examples

			For n = 2 we have 6*2^2 - 2*2 -1 = 19.

Links

Crossrefs

Cf. A098828.

Programs

  • Magma
    [a: n in [0..250] | IsPrime(a) where a is 6*n^2 - 2*n - 1]; // Vincenzo Librandi, Jul 17 2012
  • Mathematica
    Select[Table[6n^2-2n-1,{n,0,2000}],PrimeQ] (* Vincenzo Librandi, Jul 17 2012 *)
  • PARI
    for(k=1,120,if(isprime(p=6*k^2-2*k-1),print1(p,",")))

Extensions

Edited, corrected and extended by Klaus Brockhaus, Nov 12 2004

A110588 Squares of the form 2*p+3 that are squares of primes.

Original entry on oeis.org

9, 25, 49, 121, 169, 361, 529, 841, 961, 1369, 1681, 2209, 4489, 5329, 6241, 9409, 10201, 10609, 11881, 22801, 24649, 38809, 44521, 51529, 54289, 57121, 66049, 69169, 85849, 94249, 109561, 113569, 121801, 124609
Offset: 1

Views

Author

Walter Kehowski, Sep 13 2005

Keywords

Links

Crossrefs

Cf. A098828, A109358.

Programs

  • Maple
    ispower := proc(n,b) andmap(proc(w) evalb(w[2] mod b = 0) end, ifactors(n)[2]) end: a:=2: PW||a:=[]; for z from 1 to 1 do for n from 1 to 1000 do p:=ithprime(n); m:=a*p+a+1; if ispower(m,2) and isprime(sqrt(m)) then PW||a:=[op(PW||a),m] fi od; od; PW||a;
  • PARI
    isok(n) = issquare(n) && isprime(sqrtint(n)) && (type(p=(n-3)/2) == "t_INT") && isprime(p) \\ Michel Marcus, Aug 06 2013
  • PARI
    v=List();forprime(p=2,1e4,if(isprime(p^2\2-1),listput(v,p^2))); Vec(v) \\ Charles R Greathouse IV, Aug 06 2013

Extensions

More terms from Michel Marcus, Aug 06 2013

A229065 Numbers of the form 2^(p-1)+3, where p is prime.

Original entry on oeis.org

5, 7, 19, 67, 1027, 4099, 65539, 262147, 4194307, 268435459, 1073741827, 68719476739, 1099511627779, 4398046511107, 70368744177667, 4503599627370499, 288230376151711747, 1152921504606846979, 73786976294838206467, 1180591620717411303427, 4722366482869645213699
Offset: 1

Views

Author

Vincenzo Librandi, Sep 17 2013

Keywords

Comments

Primes in the sequence: 5, 7, 19, 67, 4099, 65539, 262147, 268435459, 1073741827, ...
On the other hand, for example, 2^(p-1) + 3 is composite when p == 11 (mod 12) or p == 5 (mod 18), with p>5; or when p is of the form 2*h^2+2*h*(k+2)+3*k, with k>0 and h>1.

Links

Crossrefs

Cf. A153503 (associated primes p), A098828, A057732, A057736.

Programs

  • Magma
    [2^(p-1)+3:  p in PrimesUpTo(80)];
  • Mathematica
    Table[2^(Prime[n] - 1) + 3, {n, 25}]
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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