A191554 -id:A191554 - cp's OEIS frontend

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

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

Offset: 0

Rick L. Shepherd, Jun 06 2011

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

			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)))).

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

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.

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

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

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 *)

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

;; 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

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

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

A225547 Fixed points of A225546.

Original entry on oeis.org

1, 2, 9, 12, 18, 24, 80, 108, 160, 216, 625, 720, 960, 1250, 1440, 1792, 1920, 2025, 3584, 4050, 5625, 7500, 8640, 11250, 15000, 16128, 17280, 18225, 21504, 24300, 32256, 36450, 43008, 48600, 50000, 67500, 100000, 135000, 143360, 162000, 193536, 218700, 286720, 321489, 324000, 387072, 437400, 450000, 600000

Offset: 1

Paul Tek, May 10 2013

nonn

Every number in this sequence is the product of a unique subset of A225548.
From Peter Munn, Feb 11 2020: (Start)
The terms are the numbers whose Fermi-Dirac factors (see A050376) occur symmetrically about the main diagonal of A329050.
Closed under the commutative binary operation A059897(.,.). As numbers are self-inverse under A059897, the sequence thereby forms a subgroup of the positive integers under A059897.
(End)

			The Fermi-Dirac factorization of 160 is 2 * 5 * 16. The factors 2, 5 and 16 are A329050(0,0), A329050(2,0) and A329050(0,2), having symmetry about the main diagonal of A329050. So 160 is in the sequence.

Paul Tek, Table of n, a(n) for n = 1..10000

Cf. A050376, A225546, A329050.
Closed under A059895, A059896, A059897, A306697, A329329.
Subsequences: A191554, A191555, A225548.
Cf. fixed points of the comparable A122111 involution: A088902.

A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
ff(fa) = {for (i=1, #fa~, my(p=fa[i, 1]); fa[i, 1] = A019565(fa[i, 2]); fa[i, 2] = 2^(primepi(p)-1); ); fa; } \\ A225546
pos(k, fs) = for (i=1, #fs, if (fs[i] == k, return(i)););
normalize(f) = {my(list = List()); for (k=1, #f~, my(fk = factor(f[k,1])); for (j=1, #fk~, listput(list, fk[j,1]));); my(fs = Set(list)); my(m = matrix(#fs, 2)); for (i=1, #m~, m[i,1] = fs[i]; for (k=1, #f~, m[i,2] += valuation(f[k,1], fs[i])*f[k,2];);); m;}
isok(n) = my(fa=factor(n), fb=ff(fa)); normalize(fb) == fa; \\ Michel Marcus, Aug 05 2022

A337533 1 together with nonsquares whose square part's square root is in the sequence.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68

Offset: 1

Peter Munn, Aug 31 2020

The appearance of a number is determined by its prime signature.
Every squarefree number is present, as the square root of the square part of a squarefree number is 1. Other 4th-power-free numbers are present if and only if they are nonsquare.
If the square part of nonsquarefree k is a 4th power, k does not appear.
Every positive integer k is the product of a unique subset S_k of the terms of A050376, which are arranged in array form in A329050 (primes in column 0, squares of primes in column 1, 4th powers of primes in column 2 and so on). k > 1 is in this sequence if and only if the members of S_k occur in consecutive columns of A329050, starting with column 0.
If the qualifying condition in the previous paragraph was based on the rows instead of the columns of A329050, we would get A055932. The self-inverse function defined by A225546 transposes A329050. A225546 also has multiplicative properties such that if we consider A055932 and this sequence as sets, A225546(.) maps the members of either set 1:1 onto the other set.

			4 is square and not 1, so 4 is not in the sequence.
12 = 3 * 2^2 is nonsquare, and has square part 4, whose square root (2) is in the sequence. So 12 is in the sequence.
32 = 2 * 4^2 is nonsquare, but has square part 16, whose square root (4) is not in the sequence. So 32 is not in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Square part
Index to sequences related to prime signature

Complement of A337534.
Closed under A000188(.).
A209229, A267116 are used in a formula defining this sequence.
Subsequences: A005117, A036537, A179646, A179666, A179669, A189987, A191554, A191555, A298207, A317090.
Subsequence of A164514.
A007913, A008833, A008835, A335324 give the squarefree, square and comparably related parts of a number.
Cf. A050376, A329050.
Related to A055932 via A225546.

S:= {1}:
for n from 2 to 100 do
  if not issqr(n) then
    F:= ifactors(n)[2];
    s:= mul(t[1]^floor(t[2]/2),t=F);
    if member(s,S) then S:= S union {n} fi
  fi
od:
sort(convert(S,list)); # Robert Israel, Jan 07 2025

pow2Q[n_] := n == 2^IntegerExponent[n, 2]; Select[Range[100], # == 1 || pow2Q[1 + BitOr @@ (FactorInteger[#][[;; , 2]])] &] (* Amiram Eldar, Sep 18 2020 *)

Numbers m such that A209229(A267116(m) + 1) = 1.
If A008835(a(n)) > 1 then A335324(a(n)) > 1.
If A008833(a(n)) > 1 then A007913(a(n)) > 1.

A191623 Primes of the form 1 + Product_{k=1..n} prime(k)^(2^(k-1)).

Original entry on oeis.org

3, 19, 11251, 2980024297506569894680811251

Offset: 1

Jonathan Vos Post, Jun 09 2011

nonn

Primes of the form 1 + A191554(k), associated with positions k = 1, 2, 3, and 5 there. The next one (if it exists) occurs at k >= 15 and has > 53500 digits. [Edited by R. J. Mathar, Jun 17 2011 and Joerg Arndt, Jun 21 2011]

This connects A191554 and A191555, which are deeply about primes and monic polynomial irreducible by Eisenstein's Criterion, to primes by another way, connecting additive and multiplicative number theory analogously to the relationship in Primorial primes: A014545, n such that n-th Euclid number (A006862(n)) = 1 + (Product of first n primes) is prime.

			a(1) = 1 + 2^1 = 1 + 2 = 3 is prime.
a(2) = 1 + (2^1 * 3^2) = 1 + 18 = 19 is prime.
a(3) = 1 + (2^1 * 3^2 * 5^4) = 1 + 11250 = 11251 is prime.

Cf. A000040, A191554.

Select[Array[1 + Product[Prime[k]^(2^(k - 1)), {k, #}] &, 5], PrimeQ] (* Michael De Vlieger, Feb 15 2020 *)

cp's OEIS Frontend

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

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A225547 Fixed points of A225546.

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A337533 1 together with nonsquares whose square part's square root is in the sequence.

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Maple

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A191623 Primes of the form 1 + Product_{k=1..n} prime(k)^(2^(k-1)).

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