A224705 Composite numbers n divisible by Omega(n)^2 (the square of the number of their prime factors, counted with multiplicity).

4, 16, 18, 27, 45, 63, 99, 117, 144, 153, 171, 200, 207, 216, 256, 261, 279, 300, 324, 333, 360, 369, 384, 387, 423, 450, 477, 500, 504, 531, 540, 549, 576, 603, 639, 640, 657, 675, 700, 711, 747, 750, 756, 792, 801, 873, 896, 900, 909, 927, 936, 960, 963, 981

Offset

1,1

Comments

A number n is in the sequence if and only if mod(n, A001222(n)^2) == 0 and n is not prime.

Without the restriction that n must be composite, all prime numbers would trivially be included in the sequence.

Links

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Example

a(6)=63=3*3*7, and 63 is divisible by 9=3^2; a(9)=144, which has 6 prime factors and is divisible by 36.

Maple

isA224705 := proc(n)
    if isprime(n) then
        return false;
    else
        if modp(n, numtheory[bigomega](n)^2) = 0 then
            true;
        else
            false;
        end if;
    end if;
end proc:
n := 1;
c := 4;
while n <= 10000 do
    if isA224705(c) then
        printf("%d %d\n", n, c) ;
        n := n+1 ;
    end if;
    c := c+1 ;
end do: # R. J. Mathar, Mar 14 2016

Mathematica

Select[Range[2, 1000], ! PrimeQ[#] && Mod[#, PrimeOmega[#]^2] == 0 &] (* T. D. Noe, Apr 18 2013 *)

Prog

(R) y=c(); i=2; isint<-function(x) x==as.integer(x)
while(length(y)<10000) {Omega=length(factorize(i)); if(Omega>1) if(isint(i/Omega^2)) y=c(y, i); i=i+1 }

Crossrefs

Cf. A001222, A074946, A137230, A070003, A224703.

Sequence in context: A281685 A166620 A363084 * A243298 A139719 A117102

Adjacent sequences: A224702 A224703 A224704 * A224706 A224707 A224708

Keyword

nonn

Author

Kevin L. Schwartz and Christian N. K. Anderson, Apr 16 2013

Status

approved