%I #62 Jul 01 2024 19:00:56

%S 1,4,36,900,44100,5336100,901800900,260620460100,94083986096100,

%T 49770428644836900,41856930490307832900,40224510201185827416900,

%U 55067354465423397733736100,92568222856376731590410384100,171158644061440576710668800200900

%N a(n) is the square of the product of first n primes.

%C Squares of primorials (first definition, A002110).

%C Exponential superabundant numbers: numbers k with a record value of the exponential abundancy index, A051377(k)/k > A051377(m)/m for all m < k. - _Amiram Eldar_, Apr 13 2019

%C Numbers k with a record value of A056170(k), or least number k with A056170(k) = n. - _Amiram Eldar_, Apr 15 2019

%C Empirically, these are possibly the denominators for 1 - Sum_{k=1..n} (-1)^(k+1)/prime(k)^2. The numerators are listed in A136370. - _Petros Hadjicostas_, May 14 2020

%C a(n) = least k such that rad(k/rad(k)) = A002110(n). - _David James Sycamore_, Jun 10 2024

%H Harry J. Smith, <a href="/A061742/b061742.txt">Table of n, a(n) for n = 0..100</a>

%F a(n) = Product_{j=1..n} A001248(j). - _Alois P. Heinz_, May 14 2020

%F a(n) = A228593(n) * A000040(n), for n>0. - _Marco Zárate_, Jun 11 2024

%e a(4) = 2^2 * 3^2 * 5^2 * 7^2 = 44100.

%p a:= proc(n) option remember; `if`(n=0, 1, ithprime(n)^2*a(n-1)) end:

%p seq(a(n), n=0..15); # _Alois P. Heinz_, May 14 2020

%t a[n_]:=Product[Prime[i]^2, {i, 1, n}]; (* _Vladimir Joseph Stephan Orlovsky_, Dec 05 2008 *)

%o (PARI) for(n=0,20,print1(prod(k=1,n, prime(k)^2), ", "))

%o (PARI) { n=-1; m=1; forprime (p=2, prime(101), write("b061742.txt", n++, " ", m^2); m*=p ) } \\ _Harry J. Smith_, Jul 27 2009

%o (Magma) [n eq 0 select 1 else (&*[NthPrime(j)^2: j in [1..n]]): n in [0..20]]; // _G. C. Greubel_, Apr 19 2019

%o (Sage) [product(nth_prime(j)^2 for j in (1..n)) for n in (0..20)] # _G. C. Greubel_, Apr 19 2019

%Y Equals A002110^2.

%Y Cf. A001248, A051377, A056170, A129575, A136370, A322887.

%Y Cf. A007947.

%K nonn

%O 0,2

%A _Jason Earls_, Jun 21 2001