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 1-4 of 4 results.

A173084 Semiprimes q such that q^2-4 and q^2+4 are also semiprimes.

Original entry on oeis.org

9, 21, 69, 129, 381, 2271, 3849, 3909, 3921, 5001, 5079, 5169, 5349, 7041, 16251, 18129, 18399, 20481, 22569, 22641, 22719, 22809, 28029, 32259, 35151, 38559, 39021, 39441, 39981, 41079, 42459, 48759, 48819, 49431, 50649, 61629, 67929
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 09 2010

Keywords

Comments

From Robert Israel, Jun 01 2018: (Start)
Since q^2-4 = (q-2)(q+2), for this to be a semiprime requires q-2 and q+2 to be primes.
All terms == 3 (mod 6), thus q/3 is an odd prime. (End)

Examples

			9^2-4 = 77 = 7*11 and 9^2+4 = 85 = 5*13 are semiprimes created by q=9, which adds the semiprime q=9 to the sequence.

Links

Crossrefs

Cf. A001358, A173082, A173083

Programs

  • Magma
    IsSemiprime:=func; [n: n in [4..7*10^4] | IsSemiprime(n) and IsSemiprime(n^2+4) and IsSemiprime(n^2-4)]; // Vincenzo Librandi, Jun 02 2018
  • Maple
    N:= 10^5: # to get all terms <= N
P:= select(isprime, [seq(i,i=3..N/3,2)]):
select(q -> isprime(q-2) and isprime(q+2) and numtheory:-bigomega(q^2+4)=2, 3*P); # Robert Israel, Jun 01 2018
  • Mathematica
    f[n_]:=Last/@FactorInteger[n]=={1,1}||Last/@FactorInteger[n]=={2}; lst={}; Do[If[f[n], a=n^2-4;b=n^2+4;If[f[a]&&f[b],AppendTo[lst,n]]],{n,9!}]; lst

Extensions

Definition reworded by R. J. Mathar, Mar 14 2010

A173085 Numbers n such that n, n^2 - 5, and n^2 + 5 are semiprime.

Original entry on oeis.org

26, 62, 86, 118, 134, 566, 706, 982, 1198, 1322, 1346, 1678, 1706, 1822, 2386, 2402, 2498, 2654, 2966, 3086, 3142, 3158, 3326, 3662, 4222, 4874, 5158, 5354, 5774, 6602, 6638, 6746, 6998, 7142, 7586, 7646, 7834, 8006, 8482, 8486, 8846, 9134, 9406, 10558
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 09 2010

Keywords

Comments

26^2-5=671 -> 11*61, 26^2+5=681 -> 3*227,..

Links

Crossrefs

Cf. A001358, A173082, A173083, A173084.

Programs

  • Mathematica
    f[n_]:=Last/@FactorInteger[n]=={1,1}||Last/@FactorInteger[n]=={2}; lst={};Do[If[f[n], a=n^2-5;b=n^2+5;If[f[a]&&f[b],AppendTo[lst,n]]],{n,9!}];lst
Select[Range[12000],PrimeOmega[#]==PrimeOmega[#^2-5] == PrimeOmega[ #^2+5] == 2&] (* Harvey P. Dale, Aug 29 2021 *)
  • PARI
    issemi(n)=bigomega(n)==2
is(n)=if(n%2, isprime((n^2-5)\2) && isprime((n^2+5)\2) && issemi(n), isprime(n/2) && issemi(n^2-5) && issemi(n^2+5)) \\ Charles R Greathouse IV, Sep 14 2015

Formula

a(n) >> n log n. - Charles R Greathouse IV, Sep 14 2015

Extensions

Edited by Charles R Greathouse IV, Apr 06 2010

A173086 Numbers n such that n, n^2 - 6, and n^2 + 6 are semiprime.

Original entry on oeis.org

4, 10, 49, 95, 121, 217, 247, 289, 295, 301, 305, 335, 371, 427, 469, 493, 535, 551, 581, 583, 707, 721, 745, 749, 767, 815, 817, 835, 851, 893, 899, 901, 973, 1099, 1169, 1205, 1219, 1253, 1333, 1349, 1379, 1405, 1501, 1603, 1685, 1711, 1739, 1751, 1757, 1765
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 09 2010

Keywords

Examples

			4=2*2, 4^2-6=10=2*5, 4^2+6=22=2*11, so 4 is in the sequence.

Crossrefs

Cf. A001358, A173082, A173083, A173084, A173085

Programs

  • Mathematica
    f[n_]:=Last/@FactorInteger[n]=={1,1}||Last/@FactorInteger[n]=={2}; lst={};Do[If[f[n], a=n^2-6;b=n^2+6;If[f[a]&&f[b],AppendTo[lst,n]]],{n,8!}];lst

Extensions

Edited by Charles R Greathouse IV, Apr 06 2010

A173087 Semiprimes k such that k^2 - 7 and k^2 + 7 are also semiprime.

Original entry on oeis.org

82, 142, 214, 254, 326, 358, 386, 478, 538, 542, 566, 674, 758, 802, 974, 1198, 1366, 1466, 1594, 1754, 1762, 1942, 2302, 2342, 2374, 2582, 2654, 2746, 2762, 2818, 2998, 3106, 3134, 3418, 3494, 3518, 3554, 3566, 3646, 3734, 3778, 3862, 4138, 4178, 4258
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 09 2010

Keywords

Examples

			82 = 2*41, 82^2 - 7 = 6717 = 3*2239 and 82^2 + 7 = 6731 = 53*127 are all semiprime, hence 82 is a term.

Links

Crossrefs

Cf. A001358, A173082, A173083, A173084, A173085, A173086.

Programs

  • Magma
    IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [ n: n in [2..4300] | IsSemiprime(n) and IsSemiprime(n^2-7) and IsSemiprime(n^2+7) ]; // Klaus Brockhaus, Feb 25 2010
  • Mathematica
    f[n_]:=Last/@FactorInteger[n]=={1,1}||Last/@FactorInteger[n]=={2}; lst={};Do[If[f[n], a=n^2-7;b=n^2+7;If[f[a]&&f[b],AppendTo[lst,n]]],{n,8!}];lst
Select[Range[4500],Thread[PrimeOmega[{#,#^2-7,#^2+7}]]=={2,2,2}&] (* Harvey P. Dale, Jul 27 2022 *)

Extensions

Edited by Klaus Brockhaus and N. J. A. Sloane, Feb 25 2010
Showing 1-4 of 4 results.

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

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