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

44100, 46656, 57600, 65536, 108900, 112896, 152100, 213444, 260100, 278784, 298116, 313600, 324900, 331776, 389376, 476100, 509796, 592900, 636804, 656100, 665856, 736164, 756900, 774400, 828100, 831744, 864900, 933156, 1000000, 1081600, 1218816, 1232100

Offset

1,1

Links

Table of n, a(n) for n=1..32.

Formula

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

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

Cf. A000040, A063806, A032741, A223456.

Sequence in context: A043303 A233790 A359127 * A338541 A190377 A049205

Adjacent sequences: A223454 A223455 A223456 * A223458 A223459 A223460

Keyword

nonn

Author

Christopher J. Hanson, Jul 19 2013

Status

approved