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A281140 Least k such that k is the product of n distinct primes and sigma(k) is an n-th power.

%I A281140 #57 Mar 30 2021 12:44:39
%S A281140 2,22,102,510,90510,995610,11616990,130258590,1483974030,
%T A281140 18404105922510,428454465915630,10195374973815570,240871269907008510,
%U A281140 94467020965716904490370,4580445736068712946096430,7027383957579235221501981990,419420669769073022876839238610,24967450935148397377034326845390
%N A281140 Least k such that k is the product of n distinct primes and sigma(k) is an n-th power.
%C A281140 Freiberg (Theorem 1.2) shows that there are >> (n*x^(1/n))/(log x)^(n+2) such values of k up to x. He calls the set of such numbers B*(x;+1;n). In particular, a(n) exists for each n.
%C A281140 Corresponding values of sigma(k) are 3 = 3^1, 36 = 6^2, 216 = 6^3, 1296 = 6^4, 248832 = 12^5, 2985984 = 12^6, 12^7, 12^8, 12^9, 24^10, 24^11, 24^12, 24^13, 48^14, 48^15, 60^16, 60^17, 60^18, 60^19, 84^20, 84^21, 84^22, 84^23, ...
%C A281140 a(14) <= 94467020965716904490370. - _Daniel Suteu_, Mar 28 2021
%H A281140 David A. Corneth, <a href="/A281140/b281140.txt">Table of n, a(n) for n = 1..23</a>
%H A281140 Tristan Freiberg, <a href="http://arxiv.org/abs/1008.1978">Products of shifted primes simultaneously taking perfect power values</a>, arXiv:1008.1978 [math.NT], 2010.
%e A281140 a(3) = 102 because 102 = 2 * 3 * 17 and (2 + 1)*(3 + 1)*(17 + 1) = 6^3.
%o A281140 (PARI) a(n) = my(k=2); while(!issquarefree(k) || !ispower(sigma(k), n) || omega(k)!=n, k++); k \\ _Felix Fröhlich_, Jan 17 2017
%Y A281140 Cf. A000203, A001597, A005117, A065496, A281069.
%K A281140 nonn
%O A281140 1,1
%A A281140 _Altug Alkan_, Jan 15 2017
%E A281140 a(10)-a(13) from _Jinyuan Wang_, Nov 08 2020
%E A281140 a(14) from _Daniel Suteu_ and _David A. Corneth_, Mar 28 2021
%E A281140 a(15)-a(18) from _David A. Corneth_, Mar 29 2021