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

Author

Christopher J. Hanson, Jul 19 2013

Keywords

• nonn
• easy

Examples

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

Links

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

Crossrefs

Cf. A000040, A063806, A032741.

Programs

Haskell

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

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 *)

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