A054640 a(n) is the sum of the divisors of the n-th primorial: a(n) = A000203(A002110(n)).

1, 3, 12, 72, 576, 6912, 96768, 1741824, 34836480, 836075520, 25082265600, 802632499200, 30500034969600, 1281001468723200, 56364064623820800, 2705475101943398400, 146095655504943513600, 8765739330296610816000, 543475838478389870592000, 36956357016530511200256000

Offset: 0

Author

Labos Elemer, May 15 2000

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Rafael Jakimczuk, Two Topics in Number Theory: Sum of Divisors of the Primorial and Sum of Squarefree Parts, International Mathematical Forum, Vol. 12, No. 7 (2017), pp. 331-338.

Crossrefs

Cf. A000203, A002110, A005867, A008864, A028815, A051674, A054641, A070826, A073004, A074107.

Programs

Magma

[1/2*&*[(1+NthPrime(k)): k in [0..n-1]]: n in [1..19]]; // Vincenzo Librandi, May 08 2017

Maple

a:= n-> mul(1+ithprime(j), j=1..n): seq(a(n), n=0..20); # Zerinvary Lajos, Aug 24 2008

Mathematica

Table[Product[1 + Prime[i], {i,n-1}], {n,100}] (* Geoffrey Critzer, Dec 01 2014 *)

PARI

a(n)=prod(i=1,n,prime(i)+1) \\ Charles R Greathouse IV, Feb 13 2013

SageMath

def A054640(n): return product(nth_prime(j)+1 for j in range(1,n+1))
[A054640(n) for n in range(41)] # G. C. Greubel, Aug 05 2024

Formula

a(n+1) = a(n)*(prime(n) + 1) = a(n)*A028815(n) (quotient=n-th prime+1 starting with 2).

a(n) ~ (6/Pi^2) * exp(gamma) * A002110(n) * log(prime(n)) + O(A002110(n)) (Jakimczuk, 2017). - Amiram Eldar, Feb 17 2021

a(n) = a(n-1) * A008864(n). - Flávio V. Fernandes, Mar 20 2021

a(n) = A002110(n) + A074107(n), a(n) <= A070826(1+n) [= A002110(1+n)/2] < A051674(n). - Antti Karttunen, Nov 19 2024

Extensions

a(0)=1 prepended by Alois P. Heinz, Apr 01 2021