A053576 Smallest number whose Euler totient is divisible by 2^n.

1, 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295, 8589934592, 17179869184, 34359738368, 68719476736, 137438953472, 274877906944, 549755813888, 1099511627776

Offset: 0

Author

Labos Elemer, Jan 18 2000

Keywords

• nonn

Comments

n = 32 is the first place where this differs from A001317, since 2^32 + 1 is not prime. - Mitch Harris, May 02 2007

a(8589934592) is the first unknown term; it is 2^8589934593 if F(33) = 2^(2^33)+1 is composite or F(33) otherwise. - Charles R Greathouse IV, Jul 15 2013

a(n) is the only odd element of the set phi-1(2^n), the totient inverses of 2^n. All other elements are 2*a(n), and the even elements of phi-1(2^(n-1)) * 2. - Torlach Rush, Sep 05 2017

Examples

			1,2,4,8,...,131072 divide phi of 2,3,5,15,...,196611 = 3*65537 respectively.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 0..3320

Crossrefs

Cf. A000010, A003401, A001317, A045544, A058213, A058214, A058215.

Not the same as A001317.

Programs

Mathematica

With[{s = Array[EulerPhi, 10^6]}, Table[FirstPosition[s, ?(Divisible[#, 2^n] &)][[1]], {n, 0, 19}]] (* _Michael De Vlieger, Sep 05 2017 *)

PARI

a(n)={
  if(n >= 8589934592 && valuation(n>>5,2)>27,
    warning("Result is conjectural on the nonexistence of Fermat primes >= F(33).")
  );
  if(n>31,
    return(2<Charles R Greathouse IV, Jul 15 2013

Extensions

More odd terms from Jud McCranie, Jan 25 2000