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A191555 prod(k=1..n, prime(k)^(2^(n-k))). 7
2, 12, 720, 3628800, 144850083840000, 272760108249915378892800000000, 1264767303092594444142256488682840323816161280000000000000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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 = prod(k=1..n, k^(2^(n-k))). - Jonathan Sondow, Mar 29 2014

LINKS

Table of n, a(n) for n=1..7.

FORMULA

For n > 1, a(n) = a(n-1)^2 * prime(n); a(1) = prime(1) = 2.

a(1) = 2; for n > 1, a(n) = 2^(2^(n-1)) * A003961(a(n-1)). - Antti Karttunen, Feb 06 2016

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)))).

MATHEMATICA

RecurrenceTable[{a[1] == 2, a[n] == a[n-1]^2 Prime[n]}, a, {n, 10}] (* Vincenzo Librandi, Feb 06 2016 *)

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

Cf. A191554.

Cf. also A000040, A000079, A000290, A003961, A052129, A239349, A239350, A252738, A266639.

Subsequence of A268375 and also of A267117 (apart from the initial 2).

Sequence in context: A230265 A060055 A061149 * A222207 A129933 A064320

Adjacent sequences:  A191552 A191553 A191554 * A191556 A191557 A191558

KEYWORD

nonn,easy

AUTHOR

Rick L. Shepherd, Jun 06 2011

STATUS

approved

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Last modified October 14 04:37 EDT 2019. Contains 327995 sequences. (Running on oeis4.)