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

1, 2, 8, 12, 18, 24, 36, 72, 96, 108, 128, 288, 384, 864, 972, 1152, 1944, 3456, 6144, 6912, 7776, 13122, 18432, 26244, 31104, 32768, 52488, 55296, 62208, 69984, 98304, 209952, 279936, 294912, 497664, 559872, 708588, 839808, 884736, 1679616, 3538944, 4478976

Offset: 1

Author

Reinhard Zumkeller, Jan 06 2014

Keywords

• nonn

Comments

Intersection of A007694 and A033950.

From David Morales Marciel, May 01 2015: (Start)

m is always of the form (2^i)(3^j) where i>0, j>=0.

If j=0, then m is a deficient number, and sigma(m)=2m-1. The deficiency is always 1.

If j>0, then m is an abundant number. (End)

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000010, A000005, A007694, A003586, A033950.

Programs

Haskell

a235353 n = a235353_list !! (n-1)
a235353_list = filter (\x -> mod x (a000005 x) == 0) a007694_list

Mathematica

Select[Range@ 1000000, And[Mod[#, EulerPhi@ #] == 0, Mod[#, DivisorSigma[0, #]] == 0] &] (* Michael De Vlieger, May 05 2015 *)
Select[Range[55*10^5],Mod[#,EulerPhi[#]]==Mod[#,DivisorSigma[0,#]]==0&] (* Harvey P. Dale, Feb 22 2023 *)

PARI

for(n=1,10^6,if(!(n%numdiv(n)+n%eulerphi(n)),print1(n,", "))) \\ Derek Orr, Apr 30 2015

PARI

sm3(n)=if(n<1, 0, n>>=valuation(n,2); 3^valuation(n,3)==n)
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

Python

from itertools import count, islice
from math import prod
from sympy import factorint
def A235353_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(startvalue,1)):
        f = factorint(k)
        t = prod(p**(e-1)*(p-1) for p, e in f.items())
        s = prod(e+1 for e in f.values())
        if not (k%s or k%t):
            yield k
A235353_list = list(islice(A235353_gen(),20)) # Chai Wah Wu, Mar 14 2023