A385748 Numbers k such that A384247(k) divides k.

1, 2, 6, 8, 12, 24, 32, 54, 96, 108, 128, 192, 216, 240, 384, 486, 512, 864, 972, 1536, 1728, 1944, 2048, 2160, 3072, 3456, 4374, 6000, 6144, 7776, 8192, 8748, 13824, 15552, 17496, 19440, 24576, 27648, 31104, 32768, 39366, 49152, 54000, 55296, 61440, 65280, 69984

Offset: 1

Author

Amiram Eldar, Jul 08 2025

Keywords

• nonn

Comments

(2^(2^k)-1) * 2^(2^k) is a term for k = 0..5.

Apparently, the only prime factors of any term are 2 and the Fermat primes (A019434), i.e., A092506.

Apparently, except for n = 1, a(n) / A384247(a(n)) is either 2 or 3.

Examples

			  n | a(n) | a(n) / A384247(a(n))
  --+------+---------------------
  1 |    1 | 1 / 1 = 1
  2 |    2 | 2 / 1 = 2
  3 |    6 | 6 / 2 = 3
  4 |    8 | 8 / 4 = 2
  5 |   12 | 12 / 6 = 2
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..250

Crossrefs

Cf. A019434, A092506, A384247.

Similar sequences: A007694, A298759, A319481, A335327, A373057.

Programs

Mathematica

f[p_, e_] := p^e*(1 - 1/p^(2^(IntegerExponent[e, 2]))); iphi[1] = 1; iphi[n_] := iphi[n] = Times @@ f @@@ FactorInteger[n]; q[n_] := Divisible[n, iphi[n]]; Select[Range[70000], q]

PARI

iphi(n) = {my(f = factor(n)); n * prod(i = 1, #f~, (1 - 1/f[i, 1]^(1 << valuation(f[i, 2], 2))));}
isok(k) = !( k % iphi(k));