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A342455 The fifth powers of primorials: a(n) = A002110(n)^5.

%I A342455 #32 Mar 14 2021 20:43:23
%S A342455 1,32,7776,24300000,408410100000,65774855015100000,
%T A342455 24421743243121524300000,34675383095948798128025100000,
%U A342455 85859681408495723096004822084900000,552622359415801587878908964592391520700000,11334919554709059323420895730190266747414284300000,324509123504618420438174660414872405442002404781629300000
%N A342455 The fifth powers of primorials: a(n) = A002110(n)^5.
%C A342455 The ratio G(n) = sigma(n) / (exp(gamma)*n*log(log(n))), where gamma is the Euler-Mascheroni constant (A001620), as applied to these numbers from a(1)=32 onward, develops as:
%C A342455    1: 0.8893323133
%C A342455    2: 0.7551575418
%C A342455    3: 0.7303870617
%C A342455    4: 0.7347890824
%C A342455    5: 0.7263701246
%C A342455    6: 0.7298051649
%C A342455    7: 0.7304358358
%C A342455    8: 0.7354921494
%C A342455    9: 0.7389343933
%C A342455   10: 0.7391912616
%C A342455   11: 0.7416291350
%C A342455   12: 0.7424159544
%C A342455 ...
%C A342455 Notably, after its minimum at term a(5) = 65774855015100000, it starts increasing again, albeit rather slowly. At n=10000 the ratio is 0.8632750..., and at n=40000, it is 0.87545260... Question: Does this trend continue indefinitely? In contrast, for primorials, A002110, the ratio appears to be monotonically decreasing, see comments in A342000.
%H A342455 Young Ju Choie, Nicolas Lichiardopol, Pieter Moree and Patrick Solé, <a href="https://doi.org/10.5802/jtnb.591">On Robin's criterion for the Riemann hypothesis</a>, Journal de théorie des nombres de Bordeaux, 19 no. 2 (2007), pp. 357-372.
%H A342455 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%F A342455 a(n) = A000584(A002110(n)) = A002110(n)^5.
%t A342455 FoldList[Times, 1, Prime@ Range[11]]^5 (* _Michael De Vlieger_, Mar 14 2021 *)
%o A342455 (PARI) A342455(n) = prod(i=1,n,prime(i))^5;
%o A342455 (Python)
%o A342455 from sympy.ntheory.generate import primorial
%o A342455 def A342455(n): return primorial(n)**5 if n >= 1 else 1 # _Chai Wah Wu_, Mar 13 2021
%Y A342455 Cf. A000584, A001620, A002110, A073004, A181555, A342000.
%Y A342455 Diagonal in A079474. After the initial term, also the leftmost branch in that subtree of A329886 whose root is 32.
%K A342455 nonn
%O A342455 0,2
%A A342455 _Antti Karttunen_, Mar 12 2021