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A292286 a(n) = k if the product of the divisors of n is n^k for some integer k, or -1 if no such k exists. For the ambiguous case, define a(1) = 0.

%I A292286 #33 Nov 06 2017 13:44:02
%S A292286 0,1,1,-1,1,2,1,2,-1,2,1,3,1,2,2,-1,1,3,1,3,2,2,1,4,-1,2,2,3,1,4,1,3,
%T A292286 2,2,2,-1,1,2,2,4,1,4,1,3,3,2,1,5,-1,3,2,3,1,4,2,4,2,2,1,6,1,2,3,-1,2,
%U A292286 4,1,3,2,4,1,6,1,2,3,3,2,4,1,5,-1,2,1,6,2,2,2,4,1,6,2,3,2,2,2,6,1,3,3,-1
%N A292286 a(n) = k if the product of the divisors of n is n^k for some integer k, or -1 if no such k exists. For the ambiguous case, define a(1) = 0.
%C A292286 If the number of divisors (nd) of n > 1 is odd, then a(n) = -1, else a(n) = nd/2. - _Michel Marcus_, Sep 14 2017
%C A292286 First occurrence of k beginning with -1 is A293570(r). - _Robert G. Wilson v_, Oct 10 2017
%C A292286 Records occur for A293570(r): 4, 6, 12, 24, 48, 60, 192, 240, 3072, 12288, 196608, 786432, 12582912, 805306368, etc. - _Robert G. Wilson v_, Oct 10 2017
%H A292286 Antti Karttunen, <a href="/A292286/b292286.txt">Table of n, a(n) for n = 1..20000</a>
%H A292286 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A292286 a(1) = 0, a(A000290(n+1)) = -1, a(A000037(n+1)) = A056924(A000037(n+1)), where A000290 = the squares and A000037 = the nonsquares.
%e A292286 a(10) = 2 because divisors of 10 are 1,2,5,10 with product 100 = 10^2.
%t A292286 Table[Boole[n == 1] + If[OddQ@ #, -1, #/2] &@ DivisorSigma[0, n], {n, 100}] (* _Michael De Vlieger_, Sep 15 2017 *)
%o A292286 (PARI) a(n) = if (n==1, 0, my(nd = numdiv(n)); if (nd % 2, -1, nd/2)); \\ _Michel Marcus_, Sep 14 2017
%o A292286 (PARI) a(n)=my(k=numdiv(n)); if(k%2, if(n>1, -1, 0), k/2) \\ _Charles R Greathouse IV_, Sep 19 2017
%Y A292286 Numbers n such that the product of divisors of n is n^k: A000040 (k = 1), A007422 (k = 2), A162947 (k = 3), A111398 (k = 4), A030628 (k = 5), A030630 (k = 6).
%Y A292286 Cf. A000037, A000290, A056924, A007955, A293570.
%K A292286 sign
%O A292286 1,6
%A A292286 _Juri-Stepan Gerasimov_, Sep 13 2017
%E A292286 Definition corrected by _Charles R Greathouse IV_, Sep 13 2017