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.

A226305 Numerator of Product_{d|n} b(d)^Moebius(n/d), where b() = A100371().

Original entry on oeis.org

1, 1, 3, 3, 15, 1, 63, 5, 21, 1, 1023, 5, 4095, 1, 17, 17, 65535, 1, 262143, 17, 65, 1, 4194303, 17, 69905, 1, 4161, 65, 268435455, 1, 1073741823, 257, 1025, 1, 53261, 13, 68719476735, 1, 4097, 257, 1099511627775, 1, 4398046511103, 1025, 3133, 1, 70368744177663, 257, 69810262081, 1, 65537, 4097
Offset: 1

Views

Author

N. J. A. Sloane, Jun 07 2013

Keywords

Examples

			1, 1, 3, 3, 15, 1, 63, 5, 21, 1, 1023, 5/3, 4095, 1, 17/3, 17, 65535, 1, 262143, 17/3, 65/3, 1, 4194303, 17/5, 69905, 1, 4161, 65/3, 268435455, 1, 1073741823, 257, 1025/3, 1, 53261/3, 13, ...
		

Crossrefs

Programs

  • Maple
    f:=proc(a,M) local n,b,d,t1,t2;
    b:=[];
    for n from 1 to M do
    t1:=divisors(n);
    t2:=mul(a[d]^mobius(n/d), d in t1);
    b:=[op(b),t2];
    od;
    b;
    end;
    a:=[seq(2^phi(n)-1,n=1..100)];
    f(a,100);
  • Mathematica
    Table[Numerator[Product[(2^EulerPhi[d] - 1)^MoebiusMu[n/d], {d, Divisors[n]}]], {n, 100}] (* Indranil Ghosh, Apr 14 2017 *)
  • Python
    from sympy import divisors, totient, mobius, prod
    def a(n): return prod((2**totient(d) - 1)**mobius(n//d) for d in divisors(n)).numerator
    print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Apr 14 2017

A226306 Denominator of Product_{d|n} b(d)^Moebius(n/d), where b() = A100371().

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 3, 3, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 3, 5, 1, 1, 1, 3, 1, 1, 1, 17, 1, 1, 3, 3, 1, 1, 3, 5, 3, 1, 1, 85, 1, 1, 7, 1, 15, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 17, 1, 1, 1, 325, 15, 1, 3, 5, 1, 1, 21, 3, 3, 1, 3, 257, 1, 1, 1, 1
Offset: 1

Views

Author

N. J. A. Sloane, Jun 07 2013

Keywords

Examples

			1, 1, 3, 3, 15, 1, 63, 5, 21, 1, 1023, 5/3, 4095, 1, 17/3, 17, 65535, 1, 262143, 17/3, 65/3, 1, 4194303, 17/5, 69905, 1, 4161, 65/3, 268435455, 1, 1073741823, 257, 1025/3, 1, 53261/3, 13, ...
		

Crossrefs

Programs

  • Mathematica
    Table[Denominator[Product[(2^EulerPhi[d] - 1)^MoebiusMu[n/d], {d, Divisors[n]}]], {n, 100}] (* Indranil Ghosh, Apr 14 2017 *)
  • Python
    from sympy import divisors, totient, mobius, prod
    def a(n): return prod((2**totient(d) - 1)**mobius(n//d) for d in divisors(n)).denominator
    print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Apr 14 2017

A100587 Number of nonempty subsets of divisors of n.

Original entry on oeis.org

1, 3, 3, 7, 3, 15, 3, 15, 7, 15, 3, 63, 3, 15, 15, 31, 3, 63, 3, 63, 15, 15, 3, 255, 7, 15, 15, 63, 3, 255, 3, 63, 15, 15, 15, 511, 3, 15, 15, 255, 3, 255, 3, 63, 63, 15, 3, 1023, 7, 63, 15, 63, 3, 255, 15, 255, 15, 15, 3, 4095, 3, 15, 63, 127, 15, 255, 3, 63, 15, 255, 3, 4095, 3
Offset: 1

Views

Author

Labos Elemer, Dec 01 2004

Keywords

Comments

A119347(n) <= a(n). - Reinhard Zumkeller, Jun 27 2015

Examples

			For all prime numbers p, a(p)=3, since those subsets are {{1,p},{1},{p}}.
		

Crossrefs

Programs

Formula

a(n) = -1 + 2^tau(n), where tau(n) = DivisorSigma(0, n) = A000005(n).

A243305 a(n) = 2^phi(n)+1 = A066781(n)+1.

Original entry on oeis.org

3, 3, 5, 5, 17, 5, 65, 17, 65, 17, 1025, 17, 4097, 65, 257, 257, 65537, 65, 262145, 257, 4097, 1025, 4194305, 257, 1048577, 4097, 262145, 4097, 268435457, 257, 1073741825, 65537, 1048577, 65537, 16777217, 4097, 68719476737, 262145, 16777217, 65537, 1099511627777
Offset: 1

Views

Author

Vincenzo Librandi, Jun 04 2014

Keywords

Crossrefs

Programs

  • Magma
    [2^EulerPhi(n)+1: n in [1..41]];
  • Maple
    with(numtheory); A243305:=n->2^phi(n)+1; seq(A243305(n), n=1..50); # Wesley Ivan Hurt, Jun 04 2014
  • Mathematica
    Table[2^EulerPhi[n] + 1, {n, 1, 50}]

Formula

a(n) = 2^A000010(n) + 1. - Wesley Ivan Hurt, Jun 04 2014

A243306 2^phi(n) - phi(n).

Original entry on oeis.org

1, 1, 2, 2, 12, 2, 58, 12, 58, 12, 1014, 12, 4084, 58, 248, 248, 65520, 58, 262126, 248, 4084, 1014, 4194282, 248, 1048556, 4084, 262126, 4084, 268435428, 248, 1073741794, 65520, 1048556, 65520, 16777192, 4084, 68719476700, 262126, 16777192, 65520
Offset: 1

Views

Author

Vincenzo Librandi, Jun 04 2014

Keywords

Crossrefs

Programs

  • Magma
    [2^EulerPhi(n)-EulerPhi(n): n in [1..40]];
  • Maple
    with(numtheory); A243306:=n->2^phi(n)-phi(n); seq(A243306(n), n=1..50); # Wesley Ivan Hurt, Jun 04 2014
  • Mathematica
    Table[2^EulerPhi[n] - EulerPhi[n], {n, 1, 50}]
    2^#-#&/@EulerPhi[Range[50]] (* Harvey P. Dale, Dec 19 2017 *)

Formula

a(n) = A066781(n) - A000010(n). - Wesley Ivan Hurt, Jun 04 2014

A243307 a(n) = 2^phi(n) + phi(n).

Original entry on oeis.org

3, 3, 6, 6, 20, 6, 70, 20, 70, 20, 1034, 20, 4108, 70, 264, 264, 65552, 70, 262162, 264, 4108, 1034, 4194326, 264, 1048596, 4108, 262162, 4108, 268435484, 264, 1073741854, 65552, 1048596, 65552, 16777240, 4108, 68719476772, 262162, 16777240, 65552
Offset: 1

Views

Author

Vincenzo Librandi, Jun 04 2014

Keywords

Examples

			From _Muniru A Asiru_, Jan 23 2018: (Start)
phi(1) = 1 -> a(1) = 2^1 + 1 = 3.
phi(2) = 1 -> a(2) = 2^1 + 1 = 3.
phi(7) = 6 -> a(7) = 2^6 + 6 = 70.
phi(10) = 4 -> a(10) = 2^4 + 4 = 20.
(End)
		

Crossrefs

Programs

  • GAP
    List([1..1000], n -> 2^Phi(n) + Phi(n)); # Muniru A Asiru, Jan 23 2018
  • Magma
    [2^EulerPhi(n)+EulerPhi(n): n in [1..40]];
    
  • Maple
    with(numtheory); A243307:=n->2^phi(n)+phi(n); seq(A243307(n), n=1..50); # Wesley Ivan Hurt, Jun 04 2014
  • Mathematica
    Table[2^EulerPhi[n] + EulerPhi[n], {n, 1, 50}]

Formula

a(n) = A066781(n) + A000010(n). - Wesley Ivan Hurt, Jun 04 2014
Showing 1-6 of 6 results.