OFFSET
1,1
EXAMPLE
a(1) = 44100, which has 80 divisors. 80 has 9 divisors. 9 has 2 divisors, 2 is prime. 3 steps were needed.
MATHEMATICA
d3Q[n_]:=PrimeQ[Nest[DivisorSigma[0, #]-1&, n, 3]]; Select[Range[13*10^5], d3Q] (* Harvey P. Dale, Apr 21 2016 *)
PROG
// data
uint size = Math.Power(2, 30);
uint[] divisors = new uint[size]
List<uint> A000040 = new List<uint>();
List<uint> A063806 = new List<uint>();
List<uint> A223456 = new List<uint>();
List<uint> A223457 = new List<uint>();
// calculate
for( uint i = 1; i < size; i++ )
for( uint j = i * 2; j < size; j += i )
divisors[j]++;
// assign
for( uint i = 2; i < size; i++ )
if( divisors[i] == 1 )
// A000040: Numbers with a only one proper divisor.
A000040.Add( i );
else if( divisors[divisors[i]] == 1 )
// A063806: Numbers with a prime number of proper divisors.
A063806.Add( i );
else if( divisors[divisors[divisors[i]]] == 1 )
// Numbers with a nonprime number of proper divisors
// which itself has prime number of proper divisors.
A223456.Add( i );
else if( divisors[divisors[divisors[divisors[i]]]] == 1 )
// Numbers with a nonprime number of proper divisors
// which itself has a nonprime number of proper divisors
// which itself has prime number of proper divisors.
A223457.Add( i );
else
Explode( "Conjecture is incorrect" );
CROSSREFS
KEYWORD
nonn
AUTHOR
Christopher J. Hanson, Jul 19 2013
STATUS
approved