A342455 - OEIS

A342455

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

1, 32, 7776, 24300000, 408410100000, 65774855015100000, 24421743243121524300000, 34675383095948798128025100000, 85859681408495723096004822084900000, 552622359415801587878908964592391520700000, 11334919554709059323420895730190266747414284300000, 324509123504618420438174660414872405442002404781629300000

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OFFSET

0,2

COMMENTS

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:

1: 0.8893323133

2: 0.7551575418

3: 0.7303870617

4: 0.7347890824

5: 0.7263701246

6: 0.7298051649

7: 0.7304358358

8: 0.7354921494

9: 0.7389343933

10: 0.7391912616

11: 0.7416291350

12: 0.7424159544

...

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.

LINKS

Table of n, a(n) for n=0..11.

Young Ju Choie, Nicolas Lichiardopol, Pieter Moree and Patrick Solé, On Robin's criterion for the Riemann hypothesis, Journal de théorie des nombres de Bordeaux, 19 no. 2 (2007), pp. 357-372.

Index entries for sequences related to primorial numbers

FORMULA

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

MATHEMATICA

FoldList[Times, 1, Prime@ Range[11]]^5 (* Michael De Vlieger, Mar 14 2021 *)

PROG

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

(Python)

from sympy.ntheory.generate import primorial

def A342455(n): return primorial(n)**5 if n >= 1 else 1 # Chai Wah Wu, Mar 13 2021

CROSSREFS

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