A255595 Sylvester's sequence modulo 109.

2, 3, 7, 43, 63, 92, 89, 94, 23, 71, 66, 40, 35, 101, 73, 25, 56, 29, 50, 53, 32, 12, 24, 8, 57, 32, 12, 24, 8, 57, 32, 12, 24, 8, 57, 32, 12, 24, 8, 57, 32, 12, 24, 8, 57, 32, 12, 24, 8, 57, 32, 12, 24, 8, 57, 32, 12, 24, 8, 57, 32, 12, 24, 8, 57, 32, 12, 24, 8, 57, 32, 12, 24, 8, 57, 32

Offset: 0

Author

Alonso del Arte, Mar 25 2015

Keywords

• nonn
• easy

Comments

For most small primes, it's easy to see that they have no multiples in Sylvester's sequence (A000058) by considering the sequence modulo the prime in question. For example, Sylvester's sequence modulo 41 is 2, 3, 7, 2, 3, 7, 2, 3, 7, ...

But with 109, it isn't until A000058(25) modulo 109 that we encounter the repeated value of 32. From this point forward, the period {32, 12, 24, 8, 57} is infinitely repeated. The table in Sylvester (1880) is missing the 57.

Examples

			a(4) = 43 because a(3) = 7 and 7^2 - 7 + 1 = 43.
a(5) = 63 because 43^2 - 43 + 1 = 1807 = 63 mod 109.
		

References

• J. J. Sylvester, Postscript to Note on a Point in Vulgar Fractions. American Journal of Mathematics Vol. 3, No. 4 (Dec., 1880): 389, Table.

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Programs

Mathematica

sylv109[0] := 2; sylv109[n_] := sylv109[n] = Mod[sylv109[n - 1](sylv109[n - 1] - 1) + 1, 109]; Table[sylv109[n], {n, 0, 108}]
PadRight[{2,3,7,43,63,92,89,94,23,71,66,40,35,101,73,25,56,29,50,53},120,{32,12,24,8,57}] (* Harvey P. Dale, Sep 08 2020 *)

Formula

a(0) = 2, a(n) = a(n - 1)^2 - a(n - 1) + 1 mod 109.