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A223457 Composite numbers whose number of proper divisors has a number of proper divisors which has a prime number of proper divisors. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A037147 A043303 A233790 * A190377 A049205 A234426

Adjacent sequences:  A223454 A223455 A223456 * A223458 A223459 A223460

KEYWORD

nonn

AUTHOR

Christopher J. Hanson, Jul 19 2013

STATUS

approved

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Last modified August 3 08:10 EDT 2020. Contains 336197 sequences. (Running on oeis4.)