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

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

Offset

0,2

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)

Links

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

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

Example

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)))).

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 *)

Prog

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

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.

Sequence in context: A060055 A363234 A061149 * A222207 A129933 A064320

Adjacent sequences: A191552 A191553 A191554 * A191556 A191557 A191558

Keyword

nonn,easy

Author

Rick L. Shepherd, Jun 06 2011

Extensions

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

Status

approved