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.

A066671 a(n) is the largest power of 2 that divides phi(A066669(n)).

Original entry on oeis.org

2, 2, 2, 4, 2, 2, 4, 2, 2, 4, 4, 4, 4, 4, 8, 4, 8, 8, 4, 4, 8, 2, 2, 4, 8, 4, 8, 8, 4, 2, 16, 4, 4, 8, 8, 8, 8, 8, 2, 8, 8, 8, 8, 8, 4, 2, 32, 8, 16, 16, 4, 2, 8, 16, 16, 8, 8, 2, 32, 16, 16, 8, 8, 4, 16, 4, 16, 16, 4, 8, 32, 16, 8, 16, 16, 2, 2, 16, 4, 8, 16, 4, 8, 2, 16, 8, 32, 4, 64, 32, 32
Offset: 1

Views

Author

Labos Elemer, Dec 18 2001

Keywords

Examples

			The first, 4th and 15th terms in A066669 are 7, 13 and 35; phi(7) = 2*3, phi(13) = 4*3, phi(35) = 24 = 8*3; the largest powers of 2 are 2, 4 and 8; so a(1) = 2, a(4) = 4, a(15) = 8.

Links

Crossrefs

Cf. A000010, A065966, A066669, A066670, A066672, A066673, A069177.

Programs

  • Mathematica
    Select[Array[{#1/#2, #2} & @@ {#, 2^IntegerExponent[#, 2]} &@ EulerPhi@ # &, 200], PrimeQ@ First@ # &][[All, -1]] (* Michael De Vlieger, Dec 08 2018 *)
  • PARI
    lista(nn) = {for (n=1, nn, en=eulerphi(n); if (isprime(p=en>>valuation(en, 2)), print1(en/p, ", ")););} \\ Michel Marcus, Jan 03 2017

Formula

From Amiram Eldar, Jul 18 2024: (Start)
a(n) = A069177(A066669(n)).
a(n) = 2^A066672(n). (End)

Extensions

Name corrected by Amiram Eldar, Jul 18 2024

A069256 Size of the Sylow 2-subgroup of the group GL_2(Z_n): maximal power of 2 that divides A000252(n).

Original entry on oeis.org

1, 2, 16, 32, 32, 32, 32, 512, 16, 64, 16, 512, 32, 64, 512, 8192, 512, 32, 16, 1024, 512, 32, 32, 8192, 32, 64, 16, 1024, 32, 1024, 128, 131072, 256, 1024, 1024, 512, 32, 32, 512, 16384, 128, 1024, 16, 512, 512, 64, 64, 131072, 32, 64, 8192, 1024, 32, 32, 512
Offset: 1

Views

Author

Sharon Sela (sharonsela(AT)hotmail.com), Apr 14 2002

Keywords

Links

Crossrefs

Cf. A000252, A069177.

Programs

  • Mathematica
    f[p_, e_] := 2^IntegerExponent[(p-1)*(p^2-1), 2]; f[2, e_] := 2^(4*e-3); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 02 2023 *)
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1] == 2, 1 << (4*f[i, 2]-3), 1 << valuation((f[i, 1]-1)*(f[i, 1]^2-1), 2)));} \\ Amiram Eldar, Nov 03 2023

Formula

Multiplicative with a(2^e) = 2^(4*e-3) and a(p^e) = power of 2 in prime factorization of (p - 1)*(p^2-1) for an odd prime p. - Vladeta Jovovic, Apr 17 2002

Extensions

More terms from Vladeta Jovovic, Apr 17 2002

A082506 a(n) = gcd(2^n, n - phi(n)); largest power of 2 dividing cototient(n) = A051953(n).

Original entry on oeis.org

2, 1, 1, 2, 1, 4, 1, 4, 1, 2, 1, 8, 1, 8, 1, 8, 1, 4, 1, 4, 1, 4, 1, 16, 1, 2, 1, 16, 1, 2, 1, 16, 1, 2, 1, 8, 1, 4, 1, 8, 1, 2, 1, 8, 1, 8, 1, 32, 1, 2, 1, 4, 1, 4, 1, 32, 1, 2, 1, 4, 1, 32, 1, 32, 1, 2, 1, 4, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 4, 1, 4, 1, 16, 1, 2, 1, 16, 1, 16, 1, 64, 1, 8, 1
Offset: 1

Views

Author

Labos Elemer, Apr 28 2003

Keywords

Comments

a(n)=1 if and only if n is odd or n = 2. - Robert Israel, May 31 2018

Examples

			Different from A069177, analogous sequence with totient, instead of cototient.

Links

Crossrefs

Cf. A000010, A051953, A009195, A083250, A007283, A050339, A053576, A069177.

Programs

  • Maple
    f:= n -> padic:-ordp(n - numtheory:-phi(n), 2):
map(f, [$1..100]); # Robert Israel, May 31 2018
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.