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.

A257550 Numbers n such that phi(n) = 5*phi(n+1).

Original entry on oeis.org

17907119, 18828809, 31692569, 73421039, 179467469, 322757819, 337567229, 627702389, 975314339, 2537636009, 2722271369, 3328653509, 3917646809, 5529412349, 6369847469, 11179199849, 11201693579, 11363832479, 13442120999, 16781760449, 19751331599, 20002320029
Offset: 1

Views

Author

Ray Chandler, Apr 29 2015

Keywords

Examples

			phi(17907119) = 16588800 = 5*phi(17907120).
		

Crossrefs

Programs

  • Mathematica
    a1={};nmax=10^9;last=EulerPhi[1];n=2;
    While[nRay Chandler, Apr 30 2015 *)

Extensions

a(10)-a(22) from Giovanni Resta, May 11 2015

A257865 Smallest k such that phi(k) = n*phi(k+1), where phi(n) = A000010(n) gives the value of Euler's totient function at n.

Original entry on oeis.org

1, 5, 119, 629, 17907119
Offset: 1

Views

Author

Felix Fröhlich, May 11 2015

Keywords

Comments

From Manfred Scheucher, May 27 2015: (Start)
a(6)>=3*10^8 (calculation)
a(7)>=3.5*10^13, a(8)>=4.5*10^25, a(9)>=3.0*10^47, and so on... (doubly exponential lower bound, see uploaded pdf)
239719159679 and 239742643139 admit a ratio of 5.998... and 6.008..., resp.
There might be a relation to the sequence A098026. (End)

Examples

			a(3) = 119, because phi(119) == 3*phi(120) = 96 and 119 is the smallest k where this equality holds for n = 3.
		

Crossrefs

Programs

  • Mathematica
    Table[k = 1; While[EulerPhi[k] != n EulerPhi[k + 1], k++]; k, {n, 4}] (* Michael De Vlieger, May 12 2015 *)
  • PARI
    a(n) = my(k=1); while(eulerphi(k)!=n*eulerphi(k+1), k++); k

Formula

a(n) >= exp(exp(c(n-3))) with c=exp(gamma) and gamma being the Euler-Mascheroni_constant (see pdf). - Manfred Scheucher, May 27 2015

A268126 Numbers n such that phi(n) = 4*phi(n-1).

Original entry on oeis.org

1261, 13651, 17557, 18721, 24511, 42121, 113611, 244531, 266071, 712081, 749911, 795691, 992251, 1080721, 1286731, 1458271, 1849471, 2271061, 2457691, 3295381, 3370771, 3414841, 3714751, 4061971, 4736491, 5314051, 5827081, 6566911, 6935083, 7303981, 7864081
Offset: 1

Views

Author

Jaroslav Krizek, Jan 26 2016

Keywords

Comments

See A266276(n) = the smallest numbers k such that phi(k) = n * phi(k-1) for n >=1: 2, 3, 7, 1261, 11242771, ...

Examples

			1261 is in the sequence because phi(1261) = 1152 = 4*phi(1260) = 4*288.
		

Crossrefs

Cf. A000010, A171271 (numbers n such that phi(n) = 2*phi(n-1)), A266268 (numbers n such that phi(n) = 3*phi(n-1)), A266276.
Cf. A256937.

Programs

  • Magma
    [n: n in [2..10^7] | EulerPhi(n) eq 4*EulerPhi(n-1)]
    
  • Mathematica
    Select[Range@10000000, EulerPhi@# == 4 EulerPhi[# - 1] &] (* Vincenzo Librandi, Jan 27 2016 *)
  • PARI
    isok(n) = (eulerphi(n) == 4*eulerphi(n-1)); \\ Michel Marcus, Jan 27 2016

Formula

a(n) = A172314(n) + 1. - Michel Marcus, Jan 27 2016
Showing 1-3 of 3 results.