A328701 Period in residues modulo n in iteration of x^2 + x + 1 starting at 0.
1, 1, 2, 2, 1, 2, 3, 4, 2, 1, 2, 2, 3, 3, 2, 8, 1, 2, 2, 2, 6, 2, 4, 4, 4, 3, 2, 6, 7, 2, 7, 16, 2, 1, 3, 2, 3, 2, 6, 4, 7, 6, 5, 2, 2, 4, 5, 8, 3, 4, 2, 6, 1, 2, 2, 12, 2, 7, 11, 2, 4, 7, 6, 32, 3, 2, 2, 2, 4, 3, 10, 4, 18, 3, 4, 2, 6, 6, 3, 8, 2, 7, 2, 6, 1, 5, 14, 4, 1, 2, 3
Offset: 1
Keywords
Examples
In the following example, () denotes the cycles.
A002065(n) mod 4: 0, (1, 3), so a(4) = 2.
A002065(n) mod 7: 0, (1, 3, 6), so a(7) = 3.
A002065(n) mod 29: 0, (1, 3, 13, 9, 4, 21, 28), so a(29) = 7.
A002065(n) mod 61: (0, 1, 3, 13). {A002065(n) mod 61} enters into the cycle (0, 1, 3, 13) from the very beginning, so a(61) = 0.
A002065(n) mod 64: 0, (1, 3, 13, 55, 9, 27, 53, 47, 17, 51, 29, 39, 25, 11, 5, 31, 33, 35, 45, 23, 41, 59, 21, 15, 49, 19, 61, 7, 57, 43, 37, 63), so a(64) = 32.
Programs
-
PARI
a(n) = my(v=[0],k); for(i=2, n+1, k=(v[#v]^2+v[#v]+1)%n; v=concat(v, k); for(j=1, i-1, if(v[j]==k, return(i-j))))
Formula
a(n1*n2) = lcm(a(n1),a(n2)) if gcd(n1,n2) = 1.
It seems that for e > 0, a(3^e) = 2; a(5^e) = 1 if e = 1, 4*5^(e-2) otherwise; a(7^e) = 3; a(11^e) = 2 if e = 1, 10*11^(e-2) otherwise; a(13^e) = 3 if e = 1, 12*13^(e-2) otherwise ...
Proof that a(2^e) = 2^(e-1) by induction: we will show that {f(1), f(2), ..., f(2^(e-1))} is a reduced system modulo 2^e, where f is defined in the comment section. It is easy to see that this is true for e = 1, 2.
Suppose that {f(1), f(2), ..., f(2^(e-1))} is a reduced system modulo 2^e, e = 1, 2. For each 1 <= i <= 2^(e-1), f(2^(e-1)+i) - f(i) = Sum_{j=i..2^(e-1)+i-1} (f(j+1)-f(j)) = Sum_{j=i..2^(e-1)+i-1} (f(j)^2+1) = 2^(e-1) + Sum_{j=i..2^(e-1)+i-1} f(j)^2. Of course, {f(i), f(i+1), ..., f(2^(e-1)+i-1)} is also a reduced system modulo 2^e.
Note that if x == y (mod 2^e), then x^2 == y^2 (mod 2^(e+1)). So f(2^(e-1)+i) - f(i) == 2^(e-1) + (1^2+3^2+5^2+...+(2^e-1)^2) == 2^e (mod 2^(e+1)), 1 <= i <= 2^(e-1). This shows that {f(1), f(2), ..., f(2^(e-1)), f(2^(e-1)+1), f(2^(e-1)+2), ..., f(2^e)} is a reduced system modulo 2^(e+1). QED.
Comments