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a(0) = 1; for n > 0, a(n) = 1 + 4*Product_{i=1..n-1} a(i)^2.
5

%I #25 Apr 22 2023 10:27:46

%S 1,5,101,1020101,1061522231810040101,

%T 1196154511175776540960913502483611007728163340227060101

%N a(0) = 1; for n > 0, a(n) = 1 + 4*Product_{i=1..n-1} a(i)^2.

%C Sequence designed to show that there are an infinity of primes congruent to 1 modulo 4 (A002144). Terms are not necessarily prime. Their smallest prime factors from A002144 are: 5, 101, 1020101, 53, 686743037.

%C Next term is too large to include.

%C From _Max Alekseyev_, Apr 21 2023: (Start)

%C Similarly to Sylvester's sequence (A000058), it is unknown if all terms are squarefree.

%C Primes dividing terms of this sequence are listed in A362252. Since terms are pairwise coprime, for each n prime A362252(n) divides exactly one term, whose index is A362253(n). That is, A362252(n) divides a(A362253(n)). (End)

%H S. A. Shirali, <a href="http://www.jstor.org/stable/2690862">A family portrait of primes-a case study in discrimination</a>, Math. Mag. Vol. 70, No. 4 (Oct., 1997), pp. 263-272.

%F For n > 1, a(n) = (a(n-1) - 1) * a(n-1)^2 + 1. - _Max Alekseyev_, Mar 25 2023

%o (PARI) lista(nn) = {a = vector(nn); a[1] = 5; for (n=2, nn, a[n] = 4*prod(i=1, n-1, a[i]^2) + 1;); a;}

%Y Cf. A000058, A002144, A007018, A231831, A362252, A362253.

%K nonn

%O 0,2

%A _Michel Marcus_, Nov 14 2013

%E a(0)=1 prepended by _Max Alekseyev_, Mar 25 2023