cp's OEIS Frontend

A223457 Composite numbers whose number of proper divisors has a number of proper divisors which has a prime number of proper divisors.

%I A223457 #26 Apr 21 2016 13:42:49
%S A223457 44100,46656,57600,65536,108900,112896,152100,213444,260100,278784,
%T A223457 298116,313600,324900,331776,389376,476100,509796,592900,636804,
%U A223457 656100,665856,736164,756900,774400,828100,831744,864900,933156,1000000,1081600,1218816,1232100
%N A223457 Composite numbers whose number of proper divisors has a number of proper divisors which has a prime number of proper divisors.
%F A223457 {n in A002808 : A032741(A032741(A032741(n))) is prime}.
%e A223457 a(1) = 44100, which has 80 divisors.  80 has 9 divisors. 9 has 2 divisors, 2 is prime.  3 steps were needed.
%t A223457 d3Q[n_]:=PrimeQ[Nest[DivisorSigma[0,#]-1&,n,3]]; Select[Range[13*10^5],d3Q] (* _Harvey P. Dale_, Apr 21 2016 *)
%o A223457 // data
%o A223457 uint size = Math.Power(2,30);
%o A223457 uint[] divisors = new uint[size]
%o A223457 List<uint> A000040 = new List<uint>();
%o A223457 List<uint> A063806 = new List<uint>();
%o A223457 List<uint> A223456 = new List<uint>();
%o A223457 List<uint> A223457 = new List<uint>();
%o A223457 // calculate
%o A223457 for( uint i = 1; i < size; i++ )
%o A223457     for( uint j = i * 2; j < size; j += i )
%o A223457         divisors[j]++;
%o A223457 // assign
%o A223457 for( uint i = 2; i < size; i++ )
%o A223457     if( divisors[i] == 1 )
%o A223457         // A000040: Numbers with a only one proper divisor.
%o A223457         A000040.Add( i );
%o A223457     else if( divisors[divisors[i]] == 1 )
%o A223457         // A063806: Numbers with a prime number of proper divisors.
%o A223457         A063806.Add( i );
%o A223457     else if( divisors[divisors[divisors[i]]] == 1 )
%o A223457         // Numbers with a nonprime number of proper divisors
%o A223457         // which itself has prime number of proper divisors.
%o A223457         A223456.Add( i );
%o A223457     else if( divisors[divisors[divisors[divisors[i]]]] == 1 )
%o A223457         // Numbers with a nonprime number of proper divisors
%o A223457         // which itself has a nonprime number of proper divisors
%o A223457         // which itself has prime number of proper divisors.
%o A223457         A223457.Add( i );
%o A223457     else
%o A223457         Explode( "Conjecture is incorrect" );
%Y A223457 Cf. A000040, A063806, A032741, A223456.
%K A223457 nonn
%O A223457 1,1
%A A223457 _Christopher J. Hanson_, Jul 19 2013