OFFSET
1,4
COMMENTS
Theorem 1: There are infinitely many 1's.
Proof: Suppose not. At each successive term, the sequence either grows or returns a 1. If no 1's occur after a certain term called x, then the sequence is monotonically increasing after x. It can be shown that the series grows at least as fast as the factorial between ones. Then for some sufficiently large n > x, a(n) must be greater than any term to appear in the sequence thus far. And a 1 would be returned as the next term.
Theorem 2: If a(n)=k, and k != 1, then an upper bound for the number of terms from the previous 1 to the n-th term is the length of the longest factoring of k such that the m-th factor is at least equal to m. Call this f(n). If a(n)=1, then the last 1 is at most f(a(n-1))+1 terms away.
Proof: This is because the term a(n) can be represented as x(1)*x(2)*...*x(m), with x(i) being the last time a(n-m+i-1) was seen in the sequence. Because you must cross at least the last 1 to get to the previous a(n), x(i) >= i. So the longest string of x(i)'s that can exist is one where the i-th factor is greater than or equal to i. The factors are not necessarily prime. The "longest factoring" (f(n)) refers to the longest string of numbers (x(1)*x(2)*...*x(m)) that can be multiplied to arrive at n.
Theorem 3: If a(n) is prime then a(n-2) is appearing in the sequence for the first time.
Proof: Suppose a(n) is prime. Then a(n-1) must be 1 or it must be a nontrivial divisor of a(n), unless a(n) = 1 (or the trivial case that a(n) = 0). There are no nontrivial divisors of prime numbers so a(n-1) must equal 1. It follows then that a(n-2) is appearing for the first time in the sequence because a(n-1) = 1.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log-log scatterplot of a(n) for n = 1..2^16, labeling and showing records in red, and labeling and showing first instances of differences of positions of 1's in blue.
MATHEMATICA
f[1]=1; f[n_]:=0; f2[n_]:=0; a:=(Block[{q=f2[x]}, If[q!=0, s[#]=(#-1-q)*(x), s[#]=1]])&; s[1]=1; s[2]=0; x=0; data=Reap[Sow[1]; Sow[0]; Do[Sow[x=a[n]]; f2[x]=f[x]; f[x]=n, {n, 3, 1000000}]][[2, 1]]
PROG
(Python)
from itertools import islice, count
def A350228gen():
yield from (1, 0)
b, bdict = 0, {1:(1, ), 0:(2, )}
for n in count(3):
if len(l := bdict[b]) > 1:
m = (n-1-l[-2])*b
if m in bdict:
bdict[m] = (bdict[m][-1], n)
else:
bdict[m] = (n, )
b = m
else:
bdict[1] = (bdict[1][-1], n)
b = 1
yield b
A350228_list = list(islice(A350228gen(), 20)) # Chai Wah Wu, Dec 21 2021
(PARI) findm(list, n) = {forstep (m=n-1, 1, -1, if (list[m] == list[n], return(m))); return(0); }
lista(nn) = {my(list = List([1, 0])); for (n=3, nn, my(m = findm(list, n-1)); if (m, listput(list, (n-1-m)*list[n-1]), listput(list, 1); ); ); Vec(list); } \\ Michel Marcus, Jan 16 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Jasmine Miller, Dec 20 2021
STATUS
approved