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A223456
Composite numbers whose number of proper divisors has a prime number of proper divisors.
2
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
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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved