A274205 Numbers such that the sum of divisors is twice the sum of the exponential divisors.

6, 24, 54, 216, 1638, 6552, 14256, 55860, 80262, 276822, 321048, 502740, 1107288, 1396500, 1724976, 12568500, 13564278, 20165460, 54257112, 168836850, 181489140, 504136500, 675347400, 4537228500, 28533427650, 60950102850, 114133710600, 162252212850, 243800411400, 649008851400, 734916514878

Offset: 1

Author

Paolo P. Lava, Jun 13 2016

Keywords

• nonn

Comments

All terms appear to be multiples of 6.

Subset of A011775, A069235, A175200, A215142.

a(32) > 10^12. If p*r is a term, where p is prime and r is not divisible by p, then p^3*r is also a term. - Giovanni Resta, Jun 15 2016

Examples

			Divisors of 6 are 1, 2, 3 and 6 which sum to 12. The only exponential divisor is 6. Finally 12 / 6 = 2.
Divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24 which sum to 60. Exponential divisors are 6, 24 and their sum is 30. Finally 60 / 30 = 2.
		

Crossrefs

Cf. A000203, A011775, A051377, A069235, A175200, A215142.

Programs

Maple

with(numtheory): P:=proc(q) local a,b,c,d,j,k,n,ok;
for n from 2 to q do a:=ifactors(n)[2]; b:=sort([op(divisors(n))]); c:=0;
for k from 2 to nops(b) do d:=ifactors(b[k])[2]; if nops(d)=nops(a) then
ok:=1; for j from 1 to nops(d) do if not type(a[j][2]/d[j][2],integer) then ok:=0; break; fi; od;
if ok=1 then c:=c+b[k]; fi;  fi; od; if sigma(n)=2*c  then print(n); fi; od; end: P(10^9);

Mathematica

Select[Range[10^6], 2 Times @@ Map[Sum[First[#]^d, {d, Divisors@ Last@ #}] &, FactorInteger@ #] == DivisorSigma[1, #] &] (* Michael De Vlieger, Jun 16 2016 *)

Extensions

a(16)-a(31) from Giovanni Resta, Jun 15 2016