A328701 -id:A328701 - cp's OEIS frontend

A328702 Start with 0, a(n) is the smallest number of iterations: x -> (x^2+x+1) mod n needed to run into a cycle.

0, 1, 0, 1, 2, 1, 1, 1, 2, 2, 3, 1, 0, 1, 2, 1, 3, 2, 4, 2, 1, 3, 1, 1, 2, 1, 4, 1, 1, 2, 2, 1, 3, 3, 2, 2, 2, 4, 0, 2, 4, 1, 5, 3, 2, 1, 3, 1, 3, 2, 3, 1, 9, 4, 3, 1, 4, 1, 6, 2, 0, 2, 2, 1, 2, 3, 7, 3, 1, 2, 2, 2, 3, 2, 2, 4, 3, 1, 4, 2, 4, 4, 14, 1, 3, 5, 1, 3, 8, 2, 1

Offset: 1

Jianing Song, Oct 26 2019

nonn

Let f(0) = 0, f(k+1) = (f(k)^2+f(k)+1) mod n, then a(n) is the smallest i such that f(i) = f(j) for some j > i.
Obviously a(n) <= A290731(n): f(1), f(2), ..., f(A290731(n)+1) are all of the form (s^2+s+1) mod n, so there must exists 0 <= i < j <= A290731(n)+1 such that f(i) = f(j), and a(n) <= i <= A290731(n). The equality seems to hold only for n = 2.
k divides A002065(m) for some m > 0 if and only if a(k) = 0, in which case all the indices m such that k divides A002065(m) are m = t*A328701(k), t = 0, 1, 2, 3, ...

			A002065(n) mod 4: 0, (1, 3). {A002065(n) mod 4} enters into the cycle (1, 3) from the 1st term on, so a(4) = 1.
A002065(n) mod 7: 0, (1, 3, 6). {A002065(n) mod 7} enters into the cycle (1, 3, 6) from the 1st term on, so a(7) = 1.
A002065(n) mod 11: 0, 1, 3, (2, 7). {A002065(n) mod 11} enters into the cycle (2, 7) from the 3rd term on, so a(11) = 3.
A002065(n) mod 19: 0, 1, 3, 13, (12, 5). {A002065(n) mod 19} enters into the cycle (12, 5) from the 4th term on, so a(19) = 4.
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.

Cf. A002065, A328701 (cycle length), A328703, A290731.

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(j-1))))

A328703 Numbers k dividing nonzero terms in A002065.

Original entry on oeis.org

1, 3, 13, 39, 61, 151, 169, 183, 211, 223, 453, 507, 633, 669, 739, 793, 1009, 1531, 1963, 2197, 2217, 2379, 2743, 2899, 3027, 3721, 4363, 4513, 4593, 5503, 5889, 6277, 6397, 6591, 7753, 7873, 8229, 8697, 9211, 9463, 9607, 10309, 11163, 11353, 11587, 11677, 12007, 12241, 12871

Offset: 1

Jianing Song, Oct 26 2019

nonn

k is a term if and only if A328702(k) = 0, in which case all the indices m such that k divides A002065(m) are m = t*A328701(k), t = 0, 1, 2, 3, ...

			61 divides A002065(7) = 61, so 61 is in this sequence. In addition, 61 divides A002065(m) if and only if 4 divides m.
31 is not a term: {A002065(n) mod 31} = {0, 1, 3, 13, 28, 7, 26, 21, 29, 3, 13, 28, 7, 26, 21, 29, ...}, so 31 does not divides A002065(m) for any m > 0.

Cf. A002065, A328701, A328702.

The primes in this sequence are given by A328704.

v(n) = my(v=[0],k,flag=1); 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, flag=0)); if(flag==0, break())); v
a(n) = !(v(n)[#v(n)])

cp's OEIS Frontend

A328702 Start with 0, a(n) is the smallest number of iterations: x -> (x^2+x+1) mod n needed to run into a cycle.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI