A122254 Numbers with 3-smooth Euler's totient (A000010).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 26, 27, 28, 30, 32, 34, 35, 36, 37, 38, 39, 40, 42, 45, 48, 51, 52, 54, 56, 57, 60, 63, 64, 65, 68, 70, 72, 73, 74, 76, 78, 80, 81, 84, 85, 90, 91, 95, 96, 97, 102, 104, 105, 108, 109, 111, 112, 114

Offset: 1

Author

Reinhard Zumkeller, Aug 29 2006

Keywords

• nonn
• easy

Comments

An integer n>=3 belongs to this sequence if and only if a regular n-gon can be constructed using straightedge and conic sections (details in Gibbins and Smolinsky, see below). - Austin Shapiro, Nov 14 2021

Products of 3-smooth numbers (A003586) and squarefree numbers whose prime factors are all Pierpont primes (A005109). - Amiram Eldar, Dec 03 2022

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Aliska Gibbins and Lawrence Smolinsky, Geometric Constructions with Ellipses, The Mathematical Intelligencer 31(1) (2009), 57-62.

Crossrefs

Cf. A000010, A003586 (3-smooth), A005109.

Subsequence of A122260.

Programs

Mathematica

Select[Range@115, Max[FactorInteger[EulerPhi[#]][[All, 1]]] < 5 &] (* Ivan Neretin, Jul 28 2015 *)

PARI

is(n)=n=eulerphi(n);n>>=valuation(n,2);n==3^valuation(n,3) \\ Charles R Greathouse IV, Feb 21 2013

PARI

list(lim)=my(v=List(),u,t);for(i=0,log(lim--+1.5)\log(3),t=3^i;while(t<=lim,if(isprime(t+1),listput(v,t+1));t<<=1));v=vecsort(Vec(v));u=List([1]);for(i=3,#v,for(j=1,#u,t=v[i]*u[j];if(t>lim,next(2));listput(u,t)));u=vecsort(Vec(u));v=List(u);for(i=1,#u,t=u[i];while((t*=3)<=lim,listput(v,t)));u=Vec(v);v=List(u);for(i=1,#u,t=u[i];while((t<<=1)<=lim,listput(v,t)));vecsort(Vec(v)) \\ Charles R Greathouse IV, Feb 22 2013

Python

from itertools import count, islice
from sympy import multiplicity, factorint
def A065333(n): return int(3**(multiplicity(3,m:=n>>(~n&n-1).bit_length()))==m)
def A122254_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(p<=3 or (e==1 and A065333(p-1)) for p,e in factorint(n).items()), count(max(startvalue,1)))
A122254_list = list(islice(A122254_gen(),40)) # Chai Wah Wu, Dec 20 2024

Formula

a(n) = A048135(n-2) for n>2.

a(n) = A122260(n) = A048737(n) for n < 22.

Sum_{n>=1} 1/a(n) = 3 * Product_{p > 3 in A005109} (1 + 1/p) = 5.38288865867495675807... . - Amiram Eldar, Dec 03 2022