OFFSET
1,2
COMMENTS
A rumor sequence (running modulus recurrence sequence) is defined as follows: fix integer parameters b > 1 and n > 0. Set z[0] = any integer, and, for k > 0, define z[k] to be the least nonnegative residue of b*z[k-1] modulo (k+n). The rumor sequence conjecture states that all such rumor sequences are eventually 0.
LINKS
B. Dearden, J. Iiams, and J. Metzger, Rumor Arrays, J. Integer Seq. 16 (2013), #13.9.3.
B. Dearden and J. Metzger, Running Modulus Recursions, J. Integer Seq. 13(1) (2010), #10.1.6.
FORMULA
a(n) = inf{m > 0 | z[0] = 1, z[m] = 0, and z[k] = (2*z[k-1] mod (k + n)) for k = 1..m}. - Petros Hadjicostas, Dec 13 2019
EXAMPLE
For n = 15, the z-sequence terms are 1, 2, 4, 8, 16, 12, 3, 6, 12, 0, so a(15) = 9; that is, z[0] = 1, z[1] = 2, z[2] = 4, ..., z[8] = 12, and z[9] = 0. [Edited by Petros Hadjicostas, Dec 13 2019]
MATHEMATICA
For[n=1, n<50, n++, k=0; Clear[z]; z[0]=1; z[k_]:=z[k]=Mod[2z[k-1], k+n];
While[z[k]>0, k++]; Print[k]; ]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jerry Metzger, Dec 10 2010
EXTENSIONS
More terms from Petros Hadjicostas, Dec 13 2019
STATUS
approved