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

%I A191555 #54 Jan 23 2025 22:01:31
%S A191555 1,2,12,720,3628800,144850083840000,272760108249915378892800000000,
%T A191555 1264767303092594444142256488682840323816161280000000000000000
%N A191555 a(n) = Product_{k=1..n} prime(k)^(2^(n-k)).
%C A191555 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)).
%C A191555 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
%C A191555 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
%C A191555 From _Peter Munn_, Jan 24 2020 to Feb 06 2020: (Start)
%C A191555 For n >= 0, a(n+1) is the n-th power of 12 in the monoid defined by A306697.
%C A191555 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).
%C A191555 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.
%C A191555 (End)
%C A191555 It is remarkable that 4 of the first 5 terms are factorials. - _Hal M. Switkay_, Jan 21 2025
%H A191555 Alois P. Heinz, <a href="/A191555/b191555.txt">Table of n, a(n) for n = 0..11</a>
%F A191555 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]
%F A191555 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.
%F A191555 For n > 1, a(n) = A306697(a(n-1),12) = A059896(a(n-1)^2, A003961(a(n-1))). - _Peter Munn_, Jan 24 2020
%e A191555 a(1) = 2^1 = 2 and x^2 - 2 is the minimal polynomial for the algebraic number sqrt(2).
%e A191555 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)))).
%p A191555 a:= proc(n) option remember;
%p A191555       `if`(n=0, 1, a(n-1)^2*ithprime(n))
%p A191555     end:
%p A191555 seq(a(n), n=0..8);  # _Alois P. Heinz_, Mar 05 2020
%t A191555 RecurrenceTable[{a[1] == 2, a[n] == a[n-1]^2 Prime[n]}, a, {n, 10}] (* _Vincenzo Librandi_, Feb 06 2016 *)
%t A191555 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 *)
%o A191555 (PARI) a(n) = prod(k=1, n, prime(k)^(2^(n-k)))
%o A191555 (Scheme)
%o A191555 ;; Two variants, both with memoization-macro definec.
%o A191555 (definec (A191555 n) (if (= 1 n) 2 (* (A000040 n) (A000290 (A191555 (- n 1)))))) ;; After the original recurrence.
%o A191555 (definec (A191555 n) (if (= 1 n) 2 (* (A000079 (A000079 (- n 1))) (A003961 (A191555 (- n 1)))))) ;; After the alternative recurrence - _Antti Karttunen_, Feb 06 2016
%o A191555 (Magma) [n le 1 select 2 else Self(n-1)^2*NthPrime(n): n in [1..10]]; // _Vincenzo Librandi_, Feb 06 2016
%Y A191555 Sequences with related definitions: A006939, A052129, A191554, A239350 (and thence A239349), A252738, A266639.
%Y A191555 Cf. also A000040, A000079, A003131, A072774, A329332, A331592.
%Y A191555 A000290, A003961, A059896, A306697 are used to express relationship between terms of this sequence.
%Y A191555 Subsequence of A025487, A138302, A225547, A267117 (apart from a(1) = 2), A268375, A331593.
%Y A191555 Antidiagonal products of A329050.
%K A191555 nonn,easy
%O A191555 0,2
%A A191555 _Rick L. Shepherd_, Jun 06 2011
%E A191555 a(0) added by _Peter Munn_, Feb 13 2020