A191555 a(n) = Product_{k=1..n} prime(k)^(2^(n-k)).

1, 2, 12, 720, 3628800, 144850083840000, 272760108249915378892800000000, 1264767303092594444142256488682840323816161280000000000000000

Offset: 0

Author

Rick L. Shepherd, Jun 06 2011

Keywords

• nonn
• easy

Comments

x^(2^n) - a(n) is the minimal polynomial over Q for the algebraic number sqrt(p(1)*sqrt(p(2)*...*sqrt(p(n-1)*sqrt(p(n)))...)), where p(k) is the k-th prime. Each such monic polynomial is irreducible by Eisenstein's Criterion (using p = p(n)).

A prime version of Somos's quadratic recurrence sequence A052129(n) = A052129(n-1)^2 * n = Product_{k=1..n} k^(2^(n-k)). - Jonathan Sondow, Mar 29 2014

All positive integers have unique factorizations into powers of distinct primes, and into powers of squarefree numbers with distinct exponents that are powers of 2. (See A329332 for a description of the relationship between the two.) a(n) is the least number such that both factorizations have n factors. - Peter Munn, Dec 15 2019

From Peter Munn, Jan 24 2020 to Feb 06 2020: (Start)

For n >= 0, a(n+1) is the n-th power of 12 in the monoid defined by A306697.

a(n) is the least positive integer that cannot be expressed as the product of fewer than n terms of A072774 (powers of squarefree numbers).

All terms that are less than the order of the Monster simple group (A003131) are divisors of the group's order, with a(6) exceeding its square root.

(End)

It is remarkable that 4 of the first 5 terms are factorials. - Hal M. Switkay, Jan 21 2025

Examples

			a(1) = 2^1 = 2 and x^2 - 2 is the minimal polynomial for the algebraic number sqrt(2).
a(4) = 2^8*3^4*5^2*7^1 = 3628800 and x^16 - 3628800 is the minimal polynomial for the algebraic number sqrt(2*sqrt(3*sqrt(5*sqrt(7)))).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..11

Crossrefs

Sequences with related definitions: A006939, A052129, A191554, A239350 (and thence A239349), A252738, A266639.

Cf. also A000040, A000079, A003131, A072774, A329332, A331592.

A000290, A003961, A059896, A306697 are used to express relationship between terms of this sequence.

Subsequence of A025487, A138302, A225547, A267117 (apart from a(1) = 2), A268375, A331593.

Antidiagonal products of A329050.

Programs

Magma

[n le 1 select 2 else Self(n-1)^2*NthPrime(n): n in [1..10]]; // Vincenzo Librandi, Feb 06 2016

Maple

a:= proc(n) option remember;
      `if`(n=0, 1, a(n-1)^2*ithprime(n))
    end:
seq(a(n), n=0..8);  # Alois P. Heinz, Mar 05 2020

Mathematica

RecurrenceTable[{a[1] == 2, a[n] == a[n-1]^2 Prime[n]}, a, {n, 10}] (* Vincenzo Librandi, Feb 06 2016 *)
Table[Product[Prime[k]^2^(n-k),{k,n}],{n,0,10}] (* or *) nxt[{n_,a_}]:={n+1,a^2 Prime[n+1]}; NestList[nxt,{0,1},10][[All,2]] (* Harvey P. Dale, Jan 07 2022 *)

PARI

a(n) = prod(k=1, n, prime(k)^(2^(n-k)))

Scheme

;; Two variants, both with memoization-macro definec.
(definec (A191555 n) (if (= 1 n) 2 (* (A000040 n) (A000290 (A191555 (- n 1)))))) ;; After the original recurrence.
(definec (A191555 n) (if (= 1 n) 2 (* (A000079 (A000079 (- n 1))) (A003961 (A191555 (- n 1)))))) ;; After the alternative recurrence - Antti Karttunen, Feb 06 2016

Formula

For n > 0, a(n) = a(n-1)^2 * prime(n); a(0) = 1. [edited to extend to a(0) by Peter Munn, Feb 13 2020]

a(0) = 1; for n > 0, a(n) = 2^(2^(n-1)) * A003961(a(n-1)). - Antti Karttunen, Feb 06 2016, edited Feb 13 2020 because of the new prepended starting term.

For n > 1, a(n) = A306697(a(n-1),12) = A059896(a(n-1)^2, A003961(a(n-1))). - Peter Munn, Jan 24 2020

Extensions

a(0) added by Peter Munn, Feb 13 2020