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 1-2 of 2 results.

A237040 Semiprimes of the form k^3 + 1.

Original entry on oeis.org

9, 65, 217, 4097, 5833, 10649, 21953, 74089, 195113, 216001, 343001, 373249, 474553, 1000001, 1061209, 1191017, 1404929, 3241793, 3796417, 4251529, 6859001, 9261001, 12487169, 21952001, 29791001, 35937001, 43614209, 45882713, 55742969, 62099137, 89915393, 94818817, 117649001
Offset: 1

Views

Author

Jonathan Sondow, Feb 02 2014

Keywords

Comments

k^3 + 1 is a term iff k + 1 and k^2 - k + 1 are both prime.
Is the sequence infinite? This is an analog of Landau's 4th problem, namely, are there infinitely many primes of the form k^2 + 1?
In other words: are there infinitely many primes p such that p^2 - 3*p + 3 is also prime? - Charles R Greathouse IV, Jul 02 2017

Examples

			9 = 3*3 = 2^3 + 1 is the first semiprime of the form n^3 + 1, so a(1) = 9.

Links

Crossrefs

Cf. A001358, A002383, A002496, A046315, A081256, A096173, A096174, A237037, A237038, A237039.
Cf. A242262 (semiprimes of the form k^3 - 1).

Programs

  • Magma
    IsSemiprime:= func; [s: n in [1..500] | IsSemiprime(s) where s is n^3 + 1]; // Vincenzo Librandi, Jul 02 2017
  • Mathematica
    L = Select[Range[500], PrimeQ[# + 1] && PrimeQ[#^2 - # + 1] &]; L^3 + 1
Select[Range[50]^3 + 1, PrimeOmega[#] == 2 &] (* Zak Seidov, Jun 26 2017 *)
  • PARI
    lista(nn) = for (n=1, nn, if (bigomega(sp=n^3+1) == 2, print1(sp, ", "));); \\ Michel Marcus, Jun 27 2017
  • PARI
    list(lim)=my(v=List(),n,t); forprime(p=3,sqrtnint(lim\1-1,3)+1, if(isprime(t=p^2-3*p+3), listput(v,t*p))); Vec(v) \\ Charles R Greathouse IV, Jul 02 2017

Formula

a(n) = A096173(n)^3 + 1 = 8*A237037(n)^3 + 1.

A317138 Numbers k such that (2k)^3 - 1 is a semiprime.

Original entry on oeis.org

3, 4, 6, 7, 10, 12, 19, 27, 31, 40, 45, 55, 69, 75, 82, 84, 96, 97, 136, 139, 157, 166, 174, 199, 201, 217, 250, 286, 321, 322, 360, 364, 381, 399, 406, 430, 432, 439, 460, 496, 510, 511, 535, 546, 549, 559, 565, 591, 601, 615, 630, 654, 717, 720, 724, 727, 742
Offset: 1

Views

Author

Jianing Song, Aug 01 2018

Keywords

Comments

Numbers k such that 2k - 1 and 4k^2 + 2k + 1 are both prime.

Examples

			From _K. D. Bajpai_, Nov 16 2019: (Start)
a(3) = 6 is a term because (2*6)^3 - 1 = 1727 = 11*157, which is a semiprime.
a(4) = 7 is a term because (2*7)^3 - 1 = 2743 = 13*211, which is a semiprime.
9 is not in the sequence because (2*9)^3 - 1 = 5831 = 7*7*7*17, which is not semiprime.
(End)

Links

Crossrefs

Cf. A096175, A242262.
Cf. A237037 (numbers k such that (2k)^3 + 1 is a semiprime).

Programs

  • Magma
    IsSemiprime:=func; [n: n in [2..800] | IsSemiprime(s) where s is (2*n)^3-1]; // Vincenzo Librandi, Aug 04 2018
  • Maple
    issp:= n-> not isprime(n) and numtheory[bigomega](n)=2:
select( n-> issp((2*n)^3-1),  [seq(n, n=1..200)]); # K. D. Bajpai, Nov 16 2019
  • Mathematica
    Select[Range@ 750, PrimeOmega[(2 #)^3 - 1] == 2 &] (* Michael De Vlieger, Aug 02 2018 *)
  • PARI
    for(k=1,500,if(bigomega((2*k)^3-1)==2,print1(k,", ")))

Formula

a(n) = A096175(n)/2 = (1/2)*(A242262(n) + 1)^(1/3).
Showing 1-2 of 2 results.

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

Content is available under The OEIS End-User License Agreement.