A361086 a(n) = a(n-1)*(a(n-1)^2 - 1) with a(0) = 2.

2, 6, 210, 9260790, 794226015149981778210, 500993769952171221242360491304282624082148055373236400176882790

Offset

0,1

Comments

a(6) has 189 digits so it is not displayed here.

It appears that the sequence might be squarefree, but this is unproved so far.

If n is the least index such that p^2 | a(n+1) for some prime p, then one must have p^2 | a(n)+1 or p^2 | a(n)-1, and p >= 5.

If p is prime and p^2 | a(n) for some n, then p > 10^6. - Robert Israel, Apr 16 2023

The constant c in the formula can be obtained as exp(log(a(n))/3^n) with more than 3^n/2 significant digits. - M. F. Hasler, Apr 13 2023

Links

Winston de Greef, Table of n, a(n) for n = 0..7

Dan Asimov, Interesting sequence on MathOverflow, math-fun mailing list (access restricted to subscribers), Mar 28 2023.

Fredrick M. Nelson, Does a(0)=6, a(n+1)=a(n)^3-a(n), define a square-free sequence?, MathOverflow, Mar 24 2023.

Formula

a(n) ~ c^(3^n), where c = 1.8114401993215336517307679103877676944735007990106818615618948774060863424... - Vaclav Kotesovec, Apr 13 2023

Example

After a(0) = 2, we have:
a(1) = a(0) * 3 = 6 = A002110(2),
a(2) = a(1) * 5 * 7 = 210 = A002110(4),
a(3) = a(2) * 11 * 19 * 211 = A002110(5) * 19 * 211,
a(4) = a(3) * 23 * 137 * 2939 * 101 * 91691, (*)
a(5) = a(4) * 2381 * 4547 * 73360073674487 * 4111 * 9463 * 20415865295227, (*)
a(6) = a(5) * (a(5)-1) * (a(5)+1)  where  a(5)-1 = 8297 * 2809343 * 1697219017859557 * 12663932268383565339458540396093810087  and
  a(5)+1 = 6529 * 11220511276757421196620288973 * 6838691459094922770231096737923,
etc.
(*) We list first the prime factors of a(n-1)-1, then those of a(n-1)+1.

Maple

a:= proc(n) option remember; procname(n-1)*(procname(n-1)-1)*(procname(n-1)+1) end proc:
a(0):= 2:
seq(a(i), i=0..7); # Robert Israel, Apr 16 2023

Mathematica

RecurrenceTable[{a[n] == a[n-1]*(a[n-1]^2 - 1), a[0] == 2}, a, {n, 0, 6}] (* Vaclav Kotesovec, Apr 13 2023 *)

Prog

(PARI) A361086_first(N)=vector(N, i, N=if(i>1, N*(N^2-1), 2))

Crossrefs

Cf. A002110, A007018, A228649, A361085.

Sequence in context: A091439 A285102 A285101 * A176782 A357191 A013083

Adjacent sequences: A361083 A361084 A361085 * A361087 A361088 A361089

Keyword

nonn,less

Author

M. F. Hasler, Mar 28 2023

Status

approved