

A191555


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


12




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(n1)*sqrt(p(n)))...)), where p(k) is the kth 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(n1)^2 * n = Product_{k=1..n} k^(2^(nk)).  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 nth 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(n1)^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^(n1)) * A003961(a(n1)).  Antti Karttunen, Feb 06 2016, edited Feb 13 2020 because of the new prepended starting term.
For n > 1, a(n) = A306697(a(n1),12) = A059896(a(n1)^2, A003961(a(n1))).  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(n1)^2*ithprime(n))
end:
seq(a(n), n=0..8); # Alois P. Heinz, Mar 05 2020


MATHEMATICA

RecurrenceTable[{a[1] == 2, a[n] == a[n1]^2 Prime[n]}, a, {n, 10}] (* Vincenzo Librandi, Feb 06 2016 *)
Table[Product[Prime[k]^2^(nk), {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^(nk)))
(Scheme, two variants, both with memoizationmacro 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(n1)^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: A230265 A060055 A061149 * A222207 A129933 A064320
Adjacent sequences: A191552 A191553 A191554 * A191556 A191557 A191558


KEYWORD

nonn,easy,changed


AUTHOR

Rick L. Shepherd, Jun 06 2011


EXTENSIONS

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


STATUS

approved



