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%I #18 Jun 16 2016 22:39:37
%S 6,24,54,216,1638,6552,14256,55860,80262,276822,321048,502740,1107288,
%T 1396500,1724976,12568500,13564278,20165460,54257112,168836850,
%U 181489140,504136500,675347400,4537228500,28533427650,60950102850,114133710600,162252212850,243800411400,649008851400,734916514878
%N Numbers such that the sum of divisors is twice the sum of the exponential divisors.
%C All terms appear to be multiples of 6.
%C Subset of A011775, A069235, A175200, A215142.
%C 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
%e Divisors of 6 are 1, 2, 3 and 6 which sum to 12. The only exponential divisor is 6. Finally 12 / 6 = 2.
%e 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.
%p with(numtheory): P:=proc(q) local a,b,c,d,j,k,n,ok;
%p for n from 2 to q do a:=ifactors(n)[2]; b:=sort([op(divisors(n))]); c:=0;
%p for k from 2 to nops(b) do d:=ifactors(b[k])[2]; if nops(d)=nops(a) then
%p 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;
%p 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);
%t Select[Range[10^6], 2 Times @@ Map[Sum[First[#]^d, {d, Divisors@ Last@ #}] &, FactorInteger@ #] == DivisorSigma[1, #] &] (* _Michael De Vlieger_, Jun 16 2016 *)
%Y Cf. A000203, A011775, A051377, A069235, A175200, A215142.
%K nonn
%O 1,1
%A _Paolo P. Lava_, Jun 13 2016
%E a(16)-a(31) from _Giovanni Resta_, Jun 15 2016
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