A231830 a(0) = 1; for n > 0, a(n) = 1 + 4*Product_{i=1..n-1} a(i)^2.

1, 5, 101, 1020101, 1061522231810040101, 1196154511175776540960913502483611007728163340227060101

Offset: 0

Author

Michel Marcus, Nov 14 2013

Keywords

• nonn

Comments

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.

Next term is too large to include.

From Max Alekseyev, Apr 21 2023: (Start)

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

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)

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.

Crossrefs

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

Programs

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;}

Formula

For n > 1, a(n) = (a(n-1) - 1) * a(n-1)^2 + 1. - Max Alekseyev, Mar 25 2023

Extensions

a(0)=1 prepended by Max Alekseyev, Mar 25 2023