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The fifth powers of primorials: a(n) = A002110(n)^5.
2

%I #32 Mar 14 2021 20:43:23

%S 1,32,7776,24300000,408410100000,65774855015100000,

%T 24421743243121524300000,34675383095948798128025100000,

%U 85859681408495723096004822084900000,552622359415801587878908964592391520700000,11334919554709059323420895730190266747414284300000,324509123504618420438174660414872405442002404781629300000

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

%C 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 1: 0.8893323133

%C 2: 0.7551575418

%C 3: 0.7303870617

%C 4: 0.7347890824

%C 5: 0.7263701246

%C 6: 0.7298051649

%C 7: 0.7304358358

%C 8: 0.7354921494

%C 9: 0.7389343933

%C 10: 0.7391912616

%C 11: 0.7416291350

%C 12: 0.7424159544

%C ...

%C 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 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 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>

%F a(n) = A000584(A002110(n)) = A002110(n)^5.

%t FoldList[Times, 1, Prime@ Range[11]]^5 (* _Michael De Vlieger_, Mar 14 2021 *)

%o (PARI) A342455(n) = prod(i=1,n,prime(i))^5;

%o (Python)

%o from sympy.ntheory.generate import primorial

%o def A342455(n): return primorial(n)**5 if n >= 1 else 1 # _Chai Wah Wu_, Mar 13 2021

%Y Cf. A000584, A001620, A002110, A073004, A181555, A342000.

%Y Diagonal in A079474. After the initial term, also the leftmost branch in that subtree of A329886 whose root is 32.

%K nonn

%O 0,2

%A _Antti Karttunen_, Mar 12 2021