No, probably. I think there are more sequences that are periodic mod-every-N, than sequences that obey a recurrence, with one possible glitch. a) The sequences periodic mod every N are non-denumerable, "aleph 1". Make a list of the primes & prime powers. These are the N for which the value of LCM(1,...,N) increases, and for which "mod N" has new information. We build a sequence whose period length doubles each time a new such N appears. Copy the sequence-so-far. Read the next bit of our non-denumerable random bit string; if it's a 1, add N! to each element of the copy. Proceed to the next prime-or-prime-power. b) The sequences that obey a recurrence are denumerable. We can list the recurrences, and each recurrence's starting values are a finite set of integers. This gives an enumeration of the sequences. This works as long as the sequence is determined by enough initial values of the recurrence, which is my guess at your intention. However, we can manipulate sequences from (a) to obey useless bilinear recurrences: Take any (a) sequence, insert a 0 after every term. The periodic-mod-every-N property is preserved; the periods are all doubled. The new sequence obeys the dubious recurrence s(n) s(n+1) = 0, along with your somos4^2 rule. Rich ------------- Quoting rwg@sdf.lonestar.org:
If a sequence of integers is periodic modulo every positive integer, then it obeys a linear or bilinear recurrence with constant coefficients. E.g., if s(n):= somos4(n)^2, then
- 4 s(n - 6) s(n - 1) + 29 s(n - 5) s(n - 2) + 116 s(n - 4) s(n - 3) s(n) = -------------------------------------------------------------------- s(n - 7) --rwg IN REALITY LINEARITY IMPERIALIST PRIMALITIES
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