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A235353 Numbers m such that phi(m) and tau(m) divide m, where phi = A000010 and tau = A000005.

%I A235353 #37 Jun 20 2025 17:01:44
%S A235353 1,2,8,12,18,24,36,72,96,108,128,288,384,864,972,1152,1944,3456,6144,
%T A235353 6912,7776,13122,18432,26244,31104,32768,52488,55296,62208,69984,
%U A235353 98304,209952,279936,294912,497664,559872,708588,839808,884736,1679616,3538944,4478976
%N A235353 Numbers m such that phi(m) and tau(m) divide m, where phi = A000010 and tau = A000005.
%C A235353 Intersection of A007694 and A033950.
%C A235353 From _David Morales Marciel_, May 01 2015: (Start)
%C A235353 m is always of the form (2^i)(3^j) where i>0, j>=0.
%C A235353 If j=0, then m is a deficient number, and sigma(m)=2m-1. The deficiency is always 1.
%C A235353 If j>0, then m is an abundant number. (End)
%H A235353 Reinhard Zumkeller, <a href="/A235353/b235353.txt">Table of n, a(n) for n = 1..1000</a>
%t A235353 Select[Range@ 1000000, And[Mod[#, EulerPhi@ #] == 0, Mod[#, DivisorSigma[0, #]] == 0] &] (* _Michael De Vlieger_, May 05 2015 *)
%t A235353 Select[Range[55*10^5],Mod[#,EulerPhi[#]]==Mod[#,DivisorSigma[0,#]]==0&] (* _Harvey P. Dale_, Feb 22 2023 *)
%o A235353 (Haskell)
%o A235353 a235353 n = a235353_list !! (n-1)
%o A235353 a235353_list = filter (\x -> mod x (a000005 x) == 0) a007694_list
%o A235353 (PARI) for(n=1,10^6,if(!(n%numdiv(n)+n%eulerphi(n)),print1(n,", "))) \\ _Derek Orr_, Apr 30 2015
%o A235353 (PARI) sm3(n)=if(n<1, 0, n>>=valuation(n,2); 3^valuation(n,3)==n)
%o A235353 list(lim)=my(v=List([1]),t); for(i=1,log(lim)\log(2), if(!sm3(i+1), next); for(j=0,log(lim>>i)\log(3), t=2^i*3^j; if(t%((i+1)*(j+1))==0, listput(v,t)))); Set(v) \\ _Charles R Greathouse IV_, May 05 2015
%o A235353 (Python)
%o A235353 from itertools import count, islice
%o A235353 from math import prod
%o A235353 from sympy import factorint
%o A235353 def A235353_gen(startvalue=1): # generator of terms >= startvalue
%o A235353     for k in count(max(startvalue,1)):
%o A235353         f = factorint(k)
%o A235353         t = prod(p**(e-1)*(p-1) for p, e in f.items())
%o A235353         s = prod(e+1 for e in f.values())
%o A235353         if not (k%s or k%t):
%o A235353             yield k
%o A235353 A235353_list = list(islice(A235353_gen(),20)) # _Chai Wah Wu_, Mar 14 2023
%Y A235353 Cf. A000010, A000005, A007694, A003586, A033950.
%K A235353 nonn
%O A235353 1,2
%A A235353 _Reinhard Zumkeller_, Jan 06 2014