A061742 a(n) is the square of the product of first n primes.

1, 4, 36, 900, 44100, 5336100, 901800900, 260620460100, 94083986096100, 49770428644836900, 41856930490307832900, 40224510201185827416900, 55067354465423397733736100, 92568222856376731590410384100, 171158644061440576710668800200900

Offset: 0

Author

Jason Earls, Jun 21 2001

Keywords

• nonn

Comments

Squares of primorials (first definition, A002110).

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

Numbers k with a record value of A056170(k), or least number k with A056170(k) = n. - Amiram Eldar, Apr 15 2019

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

a(n) = least k such that rad(k/rad(k)) = A002110(n). - David James Sycamore, Jun 10 2024

Examples

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

Links

• Harry J. Smith, Table of n, a(n) for n = 0..100

Crossrefs

Equals A002110^2.

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

Cf. A007947.

Programs

Magma

[n eq 0 select 1 else (&*[NthPrime(j)^2: j in [1..n]]): n in [0..20]]; // G. C. Greubel, Apr 19 2019

Maple

a:= proc(n) option remember; `if`(n=0, 1, ithprime(n)^2*a(n-1)) end:
seq(a(n), n=0..15);  # Alois P. Heinz, May 14 2020

Mathematica

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

PARI

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

PARI

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

Sage

[product(nth_prime(j)^2 for j in (1..n)) for n in (0..20)] # G. C. Greubel, Apr 19 2019

Formula

a(n) = Product_{j=1..n} A001248(j). - Alois P. Heinz, May 14 2020

a(n) = A228593(n) * A000040(n), for n>0. - Marco Zárate, Jun 11 2024