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%I #47 Aug 05 2024 16:36:10
%S 1,3,12,72,576,6912,96768,1741824,34836480,836075520,25082265600,
%T 802632499200,30500034969600,1281001468723200,56364064623820800,
%U 2705475101943398400,146095655504943513600,8765739330296610816000,543475838478389870592000,36956357016530511200256000
%N a(n) is the sum of the divisors of the n-th primorial: a(n) = A000203(A002110(n)).
%H Vincenzo Librandi, <a href="/A054640/b054640.txt">Table of n, a(n) for n = 0..200</a>
%H Rafael Jakimczuk, <a href="http://doi.org/10.12988/imf.2017.7113">Two Topics in Number Theory: Sum of Divisors of the Primorial and Sum of Squarefree Parts</a>, International Mathematical Forum, Vol. 12, No. 7 (2017), pp. 331-338.
%F a(n+1) = a(n)*(prime(n) + 1) = a(n)*A028815(n) (quotient=n-th prime+1 starting with 2).
%F a(n) ~ (6/Pi^2) * exp(gamma) * A002110(n) * log(prime(n)) + O(A002110(n)) (Jakimczuk, 2017). - _Amiram Eldar_, Feb 17 2021
%F a(n) = a(n-1) * A008864(n). - _Flávio V. Fernandes_, Mar 20 2021
%p a:= n-> mul(1+ithprime(j), j=1..n): seq(a(n), n=0..20); # _Zerinvary Lajos_, Aug 24 2008
%t Table[Product[1 + Prime[i], {i,n-1}], {n,100}] (* _Geoffrey Critzer_, Dec 01 2014 *)
%o (PARI) a(n)=prod(i=1,n,prime(i)+1) \\ _Charles R Greathouse IV_, Feb 13 2013
%o (Magma) [1/2*&*[(1+NthPrime(k)): k in [0..n-1]]: n in [1..19]]; // _Vincenzo Librandi_, May 08 2017
%o (SageMath)
%o def A054640(n): return product(nth_prime(j)+1 for j in range(1,n+1))
%o [A054640(n) for n in range(41)] # _G. C. Greubel_, Aug 05 2024
%Y Cf. A000203, A002110, A005867, A008864, A028815, A054641, A073004.
%K nonn
%O 0,2
%A _Labos Elemer_, May 15 2000
%E a(0)=1 prepended by _Alois P. Heinz_, Apr 01 2021
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