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

%I A122254 #32 Dec 20 2024 12:32:54
%S A122254 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,
%T A122254 34,35,36,37,38,39,40,42,45,48,51,52,54,56,57,60,63,64,65,68,70,72,73,
%U A122254 74,76,78,80,81,84,85,90,91,95,96,97,102,104,105,108,109,111,112,114
%N A122254 Numbers with 3-smooth Euler's totient (A000010).
%C A122254 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
%C A122254 Products of 3-smooth numbers (A003586) and squarefree numbers whose prime factors are all Pierpont primes (A005109). - _Amiram Eldar_, Dec 03 2022
%H A122254 Charles R Greathouse IV, <a href="/A122254/b122254.txt">Table of n, a(n) for n = 1..10000</a>
%H A122254 Aliska Gibbins and Lawrence Smolinsky, <a href="https://www.math.lsu.edu/~smolinsk/GeometricConstructions_5_2.pdf">Geometric Constructions with Ellipses</a>, The Mathematical Intelligencer 31(1) (2009), 57-62.
%F A122254 a(n) = A048135(n-2) for n>2.
%F A122254 a(n) = A122260(n) = A048737(n) for n < 22.
%F A122254 Sum_{n>=1} 1/a(n) = 3 * Product_{p > 3 in A005109} (1 + 1/p) = 5.38288865867495675807... . - _Amiram Eldar_, Dec 03 2022
%t A122254 Select[Range@115, Max[FactorInteger[EulerPhi[#]][[All, 1]]] < 5 &] (* _Ivan Neretin_, Jul 28 2015 *)
%o A122254 (PARI) is(n)=n=eulerphi(n);n>>=valuation(n,2);n==3^valuation(n,3) \\ _Charles R Greathouse IV_, Feb 21 2013
%o A122254 (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
%o A122254 (Python)
%o A122254 from itertools import count, islice
%o A122254 from sympy import multiplicity, factorint
%o A122254 def A065333(n): return int(3**(multiplicity(3,m:=n>>(~n&n-1).bit_length()))==m)
%o A122254 def A122254_gen(startvalue=1): # generator of terms >= startvalue
%o A122254     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)))
%o A122254 A122254_list = list(islice(A122254_gen(),40)) # _Chai Wah Wu_, Dec 20 2024
%Y A122254 Cf. A000010, A003586 (3-smooth), A005109.
%Y A122254 Subsequence of A122260.
%K A122254 nonn,easy
%O A122254 1,2
%A A122254 _Reinhard Zumkeller_, Aug 29 2006