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%I #21 Apr 08 2016 03:23:36
%S 1,2,225,4050,66528,113400,120960,92802153185280,
%T 726046074908612178739200000000000,
%U 3524292573661555639437312000000000000
%N Numbers that are equal to the product of the number of divisors of their first k powers, for some k.
%C a(2) = 2 is the only prime term possible, since the product of tau(p)^k is always even, and 2 is the only even prime. - _Michael De Vlieger_, Mar 27 2016
%C a(11) > 10^40. - _Hiroaki Yamanouchi_, Apr 07 2016
%C The corresponding k are: 1, 2, 3, 3, 3, 3, 3, 4, 5, 5. - _Michel Marcus_, Apr 08 2016
%e d(4050) * d(4050^2) = 30 * 135 = 4050;
%e d(66528) * d(66528^2) = 96 * 693 = 66528.
%p with(numtheory): P:=proc(q) local a,k,n;
%p for n from 1 to q do a:=tau(n); k:=1;
%p while a<n do k:=k+1; a:=a*tau(n^k); od;
%p if n=a then print(n); fi; od; end: P(10^6);
%t Select[Insert[Complement[Range@ #, Prime@ Range@ PrimePi@ #] &[2 10^5], 2, 2], Function[k, AnyTrue[Range@ 3, Product[DivisorSigma[0, k^i], {i, #}] == k &]]] (* _Michael De Vlieger_, Mar 25 2016 *)
%o (PARI) isok(n) = {k = 1; prd = 1; while (prd < n, prd *= numdiv(n^k); k++); prd == n;} \\ _Michel Marcus_, Apr 08 2016
%Y Cf. A000005, A270389.
%K nonn,more
%O 1,2
%A _Paolo P. Lava_, Mar 22 2016
%E a(8)-a(10) from _Hiroaki Yamanouchi_, Apr 07 2016
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