A223456 Composite numbers whose number of proper divisors has a prime number of proper divisors.

16, 36, 48, 64, 80, 81, 100, 112, 120, 144, 162, 168, 176, 196, 208, 210, 216, 225, 256, 264, 270, 272, 280, 304, 312, 324, 330, 368, 378, 384, 390, 400, 405, 408, 440, 441, 456, 462, 464, 484, 496, 510, 512, 520, 546, 552, 567, 570, 576, 592, 594, 616, 625

Offset

1,1

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

{ n: n in A002808 and A032741(A032741(n)) in A000040}.

(1 - A010051(a(n))) * A010051(a032741(a032741(a(n)))) = 1. - Reinhard Zumkeller, Sep 22 2013

Example

a(1) = 16, which has 4 proper divisors (1, 2, 4, 8). 4 has 2 proper divisors, 2 is prime. 2 steps were needed.

Maple

isA223456 := proc(n)
        local npd ;
        if not isprime(n) and n >=4 then
                npd := A032741(n) ;
                if isprime( A032741(npd)) then
                        true;
                else
                        false;
                end if ;
        else
                false;
        end if;
end proc:
for n from 16 to 630 do
        if isA223456(n) then
                printf("%d, ", n) ;
        end if;
end do: # R. J. Mathar, Sep 18 2013

Mathematica

Select[Range[1000], PrimeQ[DivisorSigma[0, DivisorSigma[0, #] - 1] - 1] &] (* Alonso del Arte, Jul 21 2013 *)

Prog

(C#)
// 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 );
(Haskell)
a223456 n = a223456_list !! (n-1)
a223456_list = filter ((== 1 ) . a010051 . a032741 . a032741) a002808_list
-- Reinhard Zumkeller, Sep 22 2013

Crossrefs

Cf. A000040, A063806, A032741.

Sequence in context: A330869 A334538 A066112 * A377563 A103843 A347747

Adjacent sequences: A223453 A223454 A223455 * A223457 A223458 A223459

Keyword

nonn,easy

Author

Christopher J. Hanson, Jul 19 2013

Status

approved