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.

A078902 Generalized Fermat primes of the form (k+1)^2^m + k^2^m, with m>1.

Original entry on oeis.org

17, 97, 257, 337, 881, 3697, 10657, 16561, 49297, 65537, 66977, 89041, 149057, 847601, 988417, 1146097, 1972097, 2070241, 2522257, 2836961, 3553777, 3959297, 4398577, 5385761, 7166897, 11073217, 17653681, 32530177, 41532497, 44048497
Offset: 1

Views

Author

T. D. Noe, Dec 12 2002

Keywords

Comments

For k=1, these are the Fermat primes A019434. Is the set of generalized Fermat primes infinite? Conjecture that there are only a finite number of generalized Fermat primes for each value of k. See A077659, which shows that in cases such as k=11, there appear to be no primes. See A078901 for generalized Fermat numbers.
See A080131 for the conjectured number of primes for each k. See A080208 for the least k such that (k+1)^2^n + k^2^n is prime. The largest probable prime of this form discovered to date is the 10217-digit 312^2^12 + 311^2^12.

Links

Crossrefs

Cf. A019434, A077659, A078900, A078901.
Cf. A080131, A080208, A019434, A078902, A080134, A153504, A152913, A194185.

Programs

  • Mathematica
    lst3=Select[lst2, PrimeQ[ # ]&] (* lst2 is from A078901 *)

A080208 a(n) is the least k such that the generalized Fermat number (k+1)^(2^n) + k^(2^n) is prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 8, 95, 31, 85, 59, 1078, 754, 311, 3508, 1828, 49957, 22844
Offset: 0

Views

Author

T. D. Noe, Feb 10 2003

Keywords

Comments

The first five terms correspond to the five known Fermat primes. The sequence A078902 lists some of the generalized Fermat primes. Bjorn and Riesel examined generalized Fermat numbers for k <= 11 and n <= 999. The sequence A080134 lists the conjectured number of primes for each k.
For n >= 10, a(n) yields a probable prime. a(13) was found by Henri Lifchitz. It is known that a(14) > 1000.

Examples

			a(5) = 8 because (k+1)^32 + k^32 is prime for k = 8 and composite for k < 8.

Links

Crossrefs

Cf. A019434, A078902, A080134, A153504, A152913, A194185, A253633.

Formula

a(n) = A253633(n) - 1.

Extensions

a(14)-a(15) from Jeppe Stig Nielsen, Nov 27 2020
a(16) by Kellen Shenton communicated by Jeppe Stig Nielsen, May 19 2023

A080131 Conjectured number of generalized Fermat primes of the form (n+1)^2^k + n^2^k, with k>1.

Original entry on oeis.org

3, 1, 2, 1, 2, 2, 1, 2, 1, 1, 0, 2, 1, 2, 0, 1
Offset: 1

Views

Author

T. D. Noe, Jan 30 2003

Keywords

Comments

Primes that are the sum of consecutive integers (k=0) and consecutive squares (k=1) are excluded. Values of k <= 16 were tested. The sequence A078902 lists some of the generalized Fermat primes. Bjorn and Riesel examined generalized Fermat numbers for n <= 11 and k <= 999.

Examples

			a(1) = 3 because there are three Fermat primes (with k>1): 17, 257, 65537.

Links

Crossrefs

Cf. A019434, A078902, A080133, A080134.

Programs

  • Mathematica
    lst={}; Do[prms=0; Do[If[PrimeQ[(n+1)^2^k+n^2^k], prms++ ], {k, 2, 16}]; AppendTo[lst, prms], {n, 16}]; lst

A081033 6th binomial transform of the periodic sequence (1,7,1,1,7,1...).

Original entry on oeis.org

1, 13, 121, 997, 7729, 57853, 423721, 3059797, 21887329, 155555053, 1100604121, 7762822597, 54632726929, 383893932253, 2694581744521, 18898693305397, 132473958606529, 928233237589453, 6502210299844921, 45538360282508197, 318882962895526129, 2232752944858526653
Offset: 0

Views

Author

Paul Barry, Mar 03 2003

Keywords

Links

Crossrefs

Cf. A080134, A080962.

Programs

  • Magma
    [4*7^n-3*5^n: n in [0..25]]; // Vincenzo Librandi, Aug 06 2013
  • Mathematica
    CoefficientList[Series[(1 + x) / ((1 - 5 x) (1 - 7 x)),{x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{12,-35},{1,13},30] (* Harvey P. Dale, Aug 18 2015 *)

Formula

a(n) = 7*a(n-1) + 6*5^(n-1).
a(n) = 4*7^n - 3*5^n.
G.f.: (1+x)/((1-5*x)*(1-7*x)). - Vincenzo Librandi, Aug 06 2013
E.g.f.: exp(5*x)*(4*exp(2*x) - 3). - Stefano Spezia, Jul 23 2024

A080133 Conjectured number of generalized Fermat primes of the form (n+1)^2^k + n^2^k, with k>0.

Original entry on oeis.org

4, 2, 2, 2, 3, 2, 2, 2, 2, 1, 0, 3, 1, 3, 0, 1
Offset: 1

Views

Author

T. D. Noe, Jan 30 2003

Keywords

Comments

Primes that are the sum of consecutive integers (k=0) are excluded. Values of k <= 16 were tested. The sequence A078902 lists some of the generalized Fermat primes. Bjorn and Riesel examined generalized Fermat numbers for n <= 11 and k <= 999.

Examples

			a(1) = 4 because there are four known Fermat primes (with k>0): 5, 17, 257, 65537.

Links

Crossrefs

Cf. A019434, A078902, A080131, A080134.

Programs

  • Mathematica
    lst={}; Do[prms=0; Do[If[PrimeQ[(n+1)^2^k+n^2^k], prms++ ], {k, 1, 16}]; AppendTo[lst, prms], {n, 16}]; lst

A122902 First occurrence of exponent n in A080121 corresponding to the minimum prime of the form (k^(2^n) + (k+1)^(2^n)) = A122900(k).

Original entry on oeis.org

1, 3, 23, 21, 10, 95, 255, 86, 59
Offset: 1

Views

Author

Alexander Adamchuk, Sep 18 2006, Oct 01 2006

Keywords

Comments

Minimum primes of the form n^(2^m) + (n+1)^(2^m) are listed in A122900. The exponents m are listed in A080121.
a(10)-a(13)>1000, a(14)-a(16)>100.

Examples

			A080121 begins with 1,1,2,1,1,2,1,2,1,5,?,1,2,1,?,2,1,?,1,?,4,1,3,1,..., where the unknown terms (denoted with ?) are at least 10. So a(1) = 1, a(2) = 3, a(3) = 23, a(4) = 21, a(5) = 10.

Links

Crossrefs

Cf. A077659, A080121, A122900, A080208, A078902, A080134, A019434, A153504, A152913, A194185.

Extensions

Edited by Max Alekseyev, Sep 09 2020
Showing 1-6 of 6 results.

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