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A231831
a(0) = 1; for n > 0, a(n) = -1 + 4*Product_{i=0..n-1} a(i)^2.
5
1, 3, 35, 44099, 85762231424099, 630794963141019085083178800095033630804099
OFFSET
0,2
COMMENTS
Sequence designed to show that there are an infinity of primes congruent to 3 modulo 4 (A002145). Terms are not necessarily prime. Their smallest prime factor from A002145 are: 3, 7, 11, 23, 4111, 2809343.
Next term is too large to include.
Similarly to Sylvester's sequence (A000058), it is unknown if all terms are squarefree (see also MathOverflow link). - Max Alekseyev, Mar 26 2023
Primes dividing terms of this sequence are listed in A362250. Since terms are pairwise coprime, for each n prime A362250(n) divides exactly one term, whose index is A362251(n). That is, A362250(n) divides a(A362251(n)). - Max Alekseyev, Apr 16 2023
LINKS
S. A. Shirali, A family portrait of primes-a case study in discrimination, Math. Mag. Vol. 70, No. 4 (Oct., 1997), pp. 263-272.
fredrickmnelson et al., Does a(0)=6, a(n+1)=a(n)^3-a(n), define a square-free sequence?, MathOverflow, 2023.
FORMULA
For n > 1, a(n) = (a(n-1) + 1) * a(n-1)^2 - 1. - Max Alekseyev, Mar 26 2023
PROG
(PARI) lista(nn) = {a = vector(nn); a[1] = 3; for (n=2, nn, a[n] = 4*prod(i=1, n-1, a[i]^2) - 1; ); a; }
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Marcus, Nov 14 2013
EXTENSIONS
a(0) = 1 prepended by Max Alekseyev, Mar 26 2023
STATUS
approved