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 1-4 of 4 results.

A066672 Exponent of the largest power of 2 that divides phi(A066669(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 3, 2, 3, 3, 2, 2, 3, 1, 1, 2, 3, 2, 3, 3, 2, 1, 4, 2, 2, 3, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3, 2, 1, 5, 3, 4, 4, 2, 1, 3, 4, 4, 3, 3, 1, 5, 4, 4, 3, 3, 2, 4, 2, 4, 4, 2, 3, 5, 4, 3, 4, 4, 1, 1, 4, 2, 3, 4, 2, 3, 1, 4, 3, 5, 2, 6, 5, 5, 4, 5, 5, 4, 3, 1, 4, 6, 5, 4, 1, 3, 4, 3
Offset: 1

Views

Author

Labos Elemer, Dec 18 2001

Keywords

Examples

			A066669(263) = 769 and phi(769) = 768 = 3*256, so a(263) = log_2(256) = log_2(phi(769)/3) = log_2(phi(A066669(263))/A066670(263)) = 8.

Links

Crossrefs

Cf. A000010, A007814, A053574, A066669, A066670, A066671.

Programs

  • Mathematica
    f[n_] := Module[{phi = EulerPhi[n], e}, e = IntegerExponent[phi, 2]; If[PrimeQ[phi/2^e], e, Nothing]]; Array[f, 300] (* Amiram Eldar, Jul 18 2024 *)

Formula

From Amiram Eldar, Jul 18 2024: (Start)
a(n) = A053574(A066669(n)).
a(n) = log_2(A066671(n)) = A007814(A000010(A066669(n))). (End)

Extensions

Name corrected by Amiram Eldar, Jul 18 2024

A066670 Primes arising in A066669: the only odd prime divisor of phi(A066669(n)).

Original entry on oeis.org

3, 3, 5, 3, 3, 3, 3, 5, 11, 5, 3, 3, 7, 5, 3, 3, 3, 5, 3, 5, 3, 11, 23, 5, 3, 13, 5, 3, 7, 29, 3, 5, 11, 3, 3, 5, 3, 5, 41, 3, 7, 5, 11, 3, 11, 23, 3, 5, 3, 3, 13, 53, 5, 3, 7, 11, 7, 29, 3, 5, 3, 5, 17, 11, 3, 23, 3, 7, 37, 5, 3, 3, 13, 5, 5, 41, 83, 3, 43, 7, 5, 29, 11, 89, 3, 11, 5, 23, 3, 3
Offset: 1

Views

Author

Labos Elemer, Dec 18 2001

Keywords

Examples

			A066669(9) = 23, phi(23) = 2*11, so a(9)=11.

Links

Crossrefs

Cf. A000010, A000265, A053575, A006530, A065966, A066669, A066671, A066672, A066673.

Programs

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

Formula

From Amiram Eldar, Jul 18 2024: (Start)
a(n) = A053575(A066669(n)).
a(n) = A000265(A000010(A066669(n))) = A006530(A000010(A066669(n))). (End)

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

A066673 Smallest term x from A066669 such that phi(x) = 2^n times some prime.

Original entry on oeis.org

7, 13, 35, 65, 97, 193, 485, 769, 1649, 3281, 8245, 12289, 24929, 49601, 124645, 197633, 423793, 786433, 2118965, 3158273, 6357089, 12648641, 31785445, 50397953, 108070513, 202113281, 540352565, 805384193, 1633771873, 3221225473, 8168859365, 12952273921
Offset: 1

Views

Author

Labos Elemer, Dec 18 2001

Keywords

Examples

			Phi of {7, 13, 35, 65, 97, 193, 485, 769} is {6, 12, 24, 48, 96, 192, 384, 768}, of which {2, 4, 8, 16, 32, 64, 128, 256} are the largest even factors.

Crossrefs

Cf. A000010, A066669-A066672.

Extensions

a(9)-a(30) from Donovan Johnson, May 23 2011
a(31)-a(32) from Donovan Johnson, May 26 2013
Showing 1-4 of 4 results.

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