Euclid–Mullin sequence

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A000945 The [lesser?] Euclid–Mullin sequence:

a (1) = 2; a (n + 1)

is the least prime factor of

1 + ∏ n k  = 1  a (k) = a (n) (a (n)  −  1) + 1, n   ≥   1

.

{2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, 2801, 11, 17, 5471, 52662739, 23003, 30693651606209, 37, 1741, 1313797957, 887, 71, 7127, 109, 23, 97, 159227, ...}

A000946 The [greater?] Euclid–Mullin sequence:

a (1) = 2; a (n + 1)

is largest prime factor of

1 + ∏ n k  = 1  a (k) = a (n) (a (n)  −  1) + 1, n   ≥   1

.

{2, 3, 7, 43, 139, 50207, 340999, 2365347734339, 4680225641471129, 1368845206580129, 889340324577880670089824574922371, ...}

Sequences

A000058 Sylvester’s sequence:

a (0) = 2; a (n + 1) = 1 + ∏ n k  = 0    a (k) = a (n) (a (n)  −  1) + 1, n   ≥   0

.

{2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, 12864938683278671740537145998360961546653259485195807, ...}

As pointed out in A000058, the sequence has a geometric interpretation relating to connected right triangles.