A235353 - OEIS

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

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OFFSET

1,2

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

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 *)

PROG

(Haskell)

a235353 n = a235353_list !! (n-1)

a235353_list = filter (\x -> mod x (a000005 x) == 0) a007694_list

(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