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%I #17 Nov 19 2012 12:35:41
%S 4,16,27,3125,3375,65536,823543,3748096,52521875,285311670611,
%T 7625597484987,302875106592253,1156831381426176,66182427701415936,
%U 827240261886336764177,511324276025564512546607,1978419655660313589123979,281633339785852578930098176
%N Numbers n for which n = (tau(n) - 1)^k with integer k.
%C tau(n) is the number of positive divisors of n.
%F Numbers n for which n = (tau(n) - 1)^k with integer k.
%e a(1) = 4 because (tau(4) - 1)^2 = (3 - 1)^2 = 4 and this is the first number satisfying this condition.
%e a(2) = 16 because (tau(16) - 1)^2 = (5 - 1)^2 = 16 and this is the second number satisfying this condition.
%e a(3) = 27 because (tau(27) - 1)^3 = (4 - 1)^3 = 27 and this is the third number satisfying this condition.
%t Select[Range[10^4], IntegerQ[Log[DivisorSigma[0, #] - 1, #]] &] (* _Alonso del Arte_, Nov 18 2012 *)
%o (PARI) v=vector(18); mx=3*10^26; c=0; for(m=2, 3440639, for(k=2, 87, n=m^k; if(n>mx, next(2)); if(m==numdiv(n)-1, c++; v[c]=n))); v=vecsort(v); for(i=1, c, print1(v[i]", ")) /* _Donovan Johnson_, Nov 19 2012 */
%Y Cf. A180936.
%K nonn
%O 1,1
%A _Zdenek Cervenka_, Nov 18 2012
%E a(10)-a(18) from _Donovan Johnson_, Nov 19 2012
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