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Composite numbers whose number of proper divisors has a number of proper divisors which has a prime number of proper divisors.
2

%I #26 Apr 21 2016 13:42:49

%S 44100,46656,57600,65536,108900,112896,152100,213444,260100,278784,

%T 298116,313600,324900,331776,389376,476100,509796,592900,636804,

%U 656100,665856,736164,756900,774400,828100,831744,864900,933156,1000000,1081600,1218816,1232100

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

%F {n in A002808 : A032741(A032741(A032741(n))) is prime}.

%e a(1) = 44100, which has 80 divisors. 80 has 9 divisors. 9 has 2 divisors, 2 is prime. 3 steps were needed.

%t d3Q[n_]:=PrimeQ[Nest[DivisorSigma[0,#]-1&,n,3]]; Select[Range[13*10^5],d3Q] (* _Harvey P. Dale_, Apr 21 2016 *)

%o // data

%o uint size = Math.Power(2,30);

%o uint[] divisors = new uint[size]

%o List<uint> A000040 = new List<uint>();

%o List<uint> A063806 = new List<uint>();

%o List<uint> A223456 = new List<uint>();

%o List<uint> A223457 = new List<uint>();

%o // calculate

%o for( uint i = 1; i < size; i++ )

%o for( uint j = i * 2; j < size; j += i )

%o divisors[j]++;

%o // assign

%o for( uint i = 2; i < size; i++ )

%o if( divisors[i] == 1 )

%o // A000040: Numbers with a only one proper divisor.

%o A000040.Add( i );

%o else if( divisors[divisors[i]] == 1 )

%o // A063806: Numbers with a prime number of proper divisors.

%o A063806.Add( i );

%o else if( divisors[divisors[divisors[i]]] == 1 )

%o // Numbers with a nonprime number of proper divisors

%o // which itself has prime number of proper divisors.

%o A223456.Add( i );

%o else if( divisors[divisors[divisors[divisors[i]]]] == 1 )

%o // Numbers with a nonprime number of proper divisors

%o // which itself has a nonprime number of proper divisors

%o // which itself has prime number of proper divisors.

%o A223457.Add( i );

%o else

%o Explode( "Conjecture is incorrect" );

%Y Cf. A000040, A063806, A032741, A223456.

%K nonn

%O 1,1

%A _Christopher J. Hanson_, Jul 19 2013