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- #!/usr/bin/ruby
- # Smallest prime divisor of A000058(n) = A007018(n) + 1 (Sylvester's sequence).
- # https://oeis.org/A323605
- # See also:
- # https://en.wikipedia.org/wiki/Sylvester%27s_sequence#Divisibility_and_factorizations
- # a(13) <= 2589377038614498251653
- # a(13) <= 2872413602289671035947763837
- # Are all the terms in Sylvester's sequence squarefree?
- func f((0)) { 1 }
- func f(n) is cached {
- f(n-1)**2 + f(n-1)
- #a(n) = a(n-1)^2 + a(n-1), a(0)=1.
- }
- #say (f(12)+1 % 2589377038614498251653)
- var t = 0
- for k in (0..30) {
- var n = f(k)+1
- say (++t, '-> ', n.trial_factor(1e8).first(-1))
- }
- __END__
- 1-> []
- 2-> []
- 3-> []
- 4-> []
- 5-> [13]
- 6-> []
- 7-> [547, 607, 1033]
- 8-> [29881, 67003, 9119521]
- 9-> []
- 10-> [181, 1987]
- 11-> [2287, 2271427]
- 12-> [73]
- 13-> []
- 14-> [52387, 5020387]
- 15-> [13999, 74203, 9638659, 57218683]
- 16-> [17881]
- 17-> [128551]
- 18-> [635263, 1286773, 21269959]
- 19-> []
- 20-> []
- 21-> [352867]
- 22-> []
- 23-> []
- 24-> [74587]
- 25-> []
- 26-> []
- 27-> [27061]
- 28-> [164299, 3229081]
- 29-> [20929]
- 30-> [1171, 298483, 97562299]
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