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 1-5 of 5 results.

A066669 Numbers m such that phi(m) = 2^k*prime for some k >= 0.

Original entry on oeis.org

7, 9, 11, 13, 14, 18, 21, 22, 23, 25, 26, 28, 29, 33, 35, 36, 39, 41, 42, 44, 45, 46, 47, 50, 52, 53, 55, 56, 58, 59, 65, 66, 69, 70, 72, 75, 78, 82, 83, 84, 87, 88, 89, 90, 92, 94, 97, 100, 104, 105, 106, 107, 110, 112, 113, 115, 116, 118, 119, 123, 130, 132, 137, 138
Offset: 1

Views

Author

Labos Elemer, Dec 18 2001

Keywords

Comments

Sequence is infinite, since 2n is in the sequence if and only if n is in the sequence. What is its density? - Charles R Greathouse IV, Feb 21 2013
Products of powers of 2, distinct terms (at least one) of A074781, and possibly (if all the factors from A074781 are Fermat primes, A019434) an additional Fermat prime (i.e., it can be divisible by a square of one Fermat prime, A330828). - Amiram Eldar, Feb 11 2025

Examples

			7 is a term because phi(7) = 6 divided by 2 is 3, a prime.
21 is a term because phi(21) = 12 divided by 4 is 3, a prime.
15 is not a term because phi(15) = 8 divided by 8 is 1, not a prime.

Links

Crossrefs

Cf. A000010, A066670, A066671, A066672, A066673, A065966.
Cf. A019434, A058500, A074781, A330828.

Programs

  • Mathematica
    Select[Range@ 138, PrimeQ@ Last@ Most@ NestWhileList[#/2 &, EulerPhi@ #, IntegerQ@ # &] &] (* Michael De Vlieger, Mar 18 2017 *)
  • PARI
    is(n)=n=eulerphi(n);isprime(n>>valuation(n,2)) \\ Charles R Greathouse IV, Feb 21 2013

A330829 Numbers of the form 2^(2*(2^n)+1)*F_n^2, where F_n is a Fermat prime A019434.

Original entry on oeis.org

72, 800, 147968, 8657174528, 36894614055915880448
Offset: 0

Views

Author

Walter Kehowski, Jan 06 2020

Keywords

Comments

Also numbers with power-spectral basis {(F_n-2)^2*F_n^2,(F_n^2-1)^2}.
The first element of the power-spectral basis of a(n) is A330830, and the second element is A330831. The first factor of a(n) is A000051(n) and the second factor is A330828.

Examples

			a(0) = 2^(2+1)*(2+1)^2 = 72, and the spectral basis is {(3-2)^2*3^2, (3^2-1)^2} = {9,64}, consisting of powers.

Crossrefs

Cf. A000215, A001146, A019434, A000051, A330828, A330830, A330831.

Programs

  • Maple
    F := proc(n) return 2^(2^n)+1 end;
G := proc(n) return 2^(2*(2^n)+1) end;
a := proc(n) if isprime(F(n)) then return G(n)*F(n)^2 fi; end;
[seq(a(n),n=0..4)];

Formula

a(n) = 2^(2*(2^n)+1)*(2^(2^n)+1)^2.

A330826 Numbers of the form 2^((2^n)+1)*F_n, where F_n is a Fermat prime, A019434.

Original entry on oeis.org

12, 40, 544, 131584, 8590065664
Offset: 1

Views

Author

Walter Kehowski, Jan 06 2020

Keywords

Comments

Also numbers with power-spectral basis {F_n^2, (F_n-1)^2}.
The first factor of a(n) is 2^A000051(n). The first element of the power-spectral basis of a(n) is A001146, and the second element is A330828.

Examples

			a(2) = 2^(2+1)*5 = 40, and the spectral basis of 40 is {25,16}, consisting of primes and powers.

Crossrefs

Cf. A001146, A000215, A019434, A000051, A330828.

Programs

  • Maple
    F := n -> 2^(2^n)+1;
a := proc(n) if isprime(F(n)) then return 2^((2^n)+1)*F(n) fi; end;

Formula

a(n) =2^A000051(n)*A019434(n).

A330830 Numbers of the form (F_n-2)^2*F_n^2, where F_n is a Fermat prime, A019434. Also the first element of the power-spectral basis of A330829.

Original entry on oeis.org

9, 225, 65025, 4294836225, 18446744065119617025
Offset: 1

Views

Author

Walter Kehowski, Jan 06 2020

Keywords

Comments

The second element of the power-spectral basis of A330829 is A330831. We also have a(n) = (2^(2*2^n)-1)^2.

Examples

			a(1) = (3-2)^2*3^2  =9.

Crossrefs

Cf. A000215, A001146, A019434, A330828.

Programs

  • Maple
    a := proc(n) if isprime(2^(2^n)+1) then return (2^(2*2^n)-1)^2 fi; end;
[seq(a(n),n=0..4)];

Formula

a(n) = (A019434(n)-2)^2*A019434(n)^2.

A330831 a(n) = (F_n^2 - 1)^2, where F_n is a Fermat prime, A019434.

Original entry on oeis.org

64, 576, 82944, 4362338304, 18447869990796263424
Offset: 1

Views

Author

Walter Kehowski, Jan 06 2020

Keywords

Comments

Also the second element of the power-spectral basis of A330829.
The first element of the power-spectral basis of A330829 is A330830.

Examples

			a(0) = (3^2 - 1)^2 = 64.

Crossrefs

Cf. A000215, A001146, A019434, A330827, A330828, A330829, A330830.

Programs

  • Maple
    F := proc(n) return 2^(2^n)+1 end;
a := proc(n) if isprime(F(n)) then return (F(n)^2-1)^2 fi; end;
[seq(a(n),n=0..4)];

Formula

a(n) = (F(n)^2 - 1)^2, where F(n) = 2^(2^n)+1 is a Fermat prime.
Showing 1-5 of 5 results.

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