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A349006 a(1) = 1; for n > 1, a(n) is the smallest number m such that sigma(m) = tau(m)^n or 0 if no such m exists.

%I A349006 #8 Sep 08 2022 08:46:26
%S A349006 1,3,7,217,31,3937,127,57337,253921,917497,3670009,16252897,8191,
%T A349006 61079603913818329,1073602561,4294434817,131071,66571993057,524287,
%U A349006 1208766717309082486038529,9222228542614937599,17590038552577,500367932999371587367,281472829095937,1125897758834689
%N A349006 a(1) = 1; for n > 1, a(n) is the smallest number m such that sigma(m) = tau(m)^n or 0 if no such m exists.
%C A349006 See A051281 for numbers m such that sigma(m) = tau(m)^k where k = integer.
%C A349006 a(n) = 0 for n = 76, 81, ...
%e A349006 a(4) = 217 because 217 is the smallest number m such that sigma(m) = tau(m)^4; sigma(217) = 256 = tau(217)^4  = 4^4.
%t A349006 Table[Block[{m = n}, While[#2 != #1^n & @@ DivisorSigma[{0, 1}, m], m++]; m], {n, 10}] (* _Michael De Vlieger_, Nov 05 2021 *)
%o A349006 (Magma) [1] cat [Min([m: m in[2..10^6] | &+Divisors(m) eq #Divisors(m)^n]): n in [2..10]]
%Y A349006 Cf. A000005 (tau), A000203 (sigma), A051281, A334455, A349007.
%K A349006 nonn
%O A349006 1,2
%A A349006 _Jaroslav Krizek_, Nov 05 2021