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

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

Offset: 0

Author

Antti Karttunen, Mar 12 2021

Keywords

• nonn

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

• 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

Crossrefs

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

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

Programs

Mathematica

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

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

Formula

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