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 1-3 of 3 results.

A108814 Numbers k such that k^4 + 4 is semiprime.

Original entry on oeis.org

3, 5, 15, 25, 55, 125, 205, 385, 465, 635, 645, 715, 1095, 1145, 1175, 1245, 1275, 1315, 1375, 1565, 1615, 1675, 1685, 1965, 2055, 2085, 2095, 2405, 2455, 2535, 2665, 2835, 2925, 3135, 3305, 3535, 3755, 3775, 4025, 4155, 4175, 4365, 4605, 4615, 4735, 4785
Offset: 1

Views

Author

Jason Earls, Jul 10 2005

Keywords

Comments

Except for the first, all the terms above generate brilliant numbers.
Numbers n such that n - 1 + i and n + 1 + i are (twin) Gaussian primes, see Shanks. - Charles R Greathouse IV, Apr 20 2011

Links

Crossrefs

Cf. A001358, A057781, A096012.

Programs

  • Magma
    IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [ n: n in [1..5000] | IsSemiprime(n^4+4)]; // Vincenzo Librandi, Apr 20 2011
  • Mathematica
    Select[Range[5000],PrimeOmega[#^4+4]==2&] (* Harvey P. Dale, Sep 07 2017 *)
  • PARI
    forstep(n=1,1e5,2,if(isprime(n^2-2*n+2) && isprime(n^2+2*n+2), print1(n", "))) \\ Charles R Greathouse IV, Apr 20 2011

Formula

a(k) = A096012(k) + 1. (Because n^4+4 = ((n-1)^2+1)((n+1)^2+1).) - Jeppe Stig Nielsen, Feb 26 2016

A272298 a(n) = n^4 + 324.

Original entry on oeis.org

324, 325, 340, 405, 580, 949, 1620, 2725, 4420, 6885, 10324, 14965, 21060, 28885, 38740, 50949, 65860, 83845, 105300, 130645, 160324, 194805, 234580, 280165, 332100, 390949, 457300, 531765, 614980, 707605, 810324, 923845, 1048900, 1186245, 1336660, 1500949, 1679940, 1874485, 2085460
Offset: 0

Views

Author

Bruno Berselli, Apr 25 2016

Keywords

Comments

This is the case k=3 of Sophie Germain's Identity n^4+(2*k^2)^2 = ((n-k)^2+k^2)*((n+k)^2+k^2).

Links

Crossrefs

Cf. A005917.
Subsequence of A227855.
Cf. A000583 (k=0), A057781 (k=1), A272297 (k=2).

Programs

  • Magma
    [n^4+324: n in [0..40]];
  • Mathematica
    Table[n^4 + 324, {n, 0, 40}]
LinearRecurrence[{5,-10,10,-5,1},{324,325,340,405,580},40] (* Harvey P. Dale, Jan 20 2021 *)
  • Maxima
    makelist(n^4+324, n, 0, 40);
  • PARI
    vector(40, n, n--; n^4+324)
  • Python
    [n**4+324 for n in range(40)]
  • Python
    for n in range(0, 10**5):print(n**4+324,end=", ") # Soumil Mandal, Apr 30 2016
  • Sage
    [n^4+324 for n in (0..40)]

Formula

O.g.f.: (324 - 1295*x + 1955*x^2 - 1285*x^3 + 325*x^4)/(1 - x)^5. [Corrected by Georg Fischer, May 23 2019]
E.g.f.: (324 + x + 7*x^2 + 6*x^3 + x^4)*exp(x).
a(n) = (n^2 - 18)^2 + (6*n)^2.

A272297 a(n) = n^4 + 64.

Original entry on oeis.org

64, 65, 80, 145, 320, 689, 1360, 2465, 4160, 6625, 10064, 14705, 20800, 28625, 38480, 50689, 65600, 83585, 105040, 130385, 160064, 194545, 234320, 279905, 331840, 390689, 457040, 531505, 614720, 707345, 810064, 923585, 1048640, 1185985, 1336400, 1500689, 1679680, 1874225, 2085200
Offset: 0

Views

Author

Bruno Berselli, Apr 25 2016

Keywords

Comments

This is the case k=2 of Sophie Germain's Identity n^4+(2*k^2)^2 = ((n-k)^2+k^2)*((n+k)^2+k^2).

Links

Crossrefs

Cf. A005917.
Subsequence of A227855.
Cf. A000583 (k=0), A057781 (k=1), A272298 (k=3).

Programs

  • Magma
    [n^4+64: n in [0..40]];
  • Mathematica
    Table[n^4 + 64, {n, 0, 40}]
  • Maxima
    makelist(n^4+64, n, 0, 40);
  • PARI
    vector(40, n, n--; n^4+64)
  • Python
    [n**4+64 for n in range(40)]
  • Python
    for n in range(0,10**5):print(n**4+64) # Soumil Mandal, Apr 30 2016
  • Sage
    [n^4+64 for n in (0..40)]

Formula

O.g.f.: (64 - 255*x + 395*x^2 - 245*x^3 + 65*x^4)/(1 - x)^5.
E.g.f.: (64 + x + 7*x^2 + 6*x^3 + x^4)*exp(x).
a(n) = (n^2 - 8)^2 + (4*n)^2.
Showing 1-3 of 3 results.

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