A381019 a(n) is the smallest positive integer not yet in the sequence such that a(n) is relatively prime to a(n-i) for all 1 <= i <= min(a(n), n-1).
1, 2, 3, 5, 7, 11, 4, 13, 17, 19, 23, 29, 9, 31, 37, 8, 41, 43, 47, 53, 59, 61, 6, 67, 71, 73, 79, 83, 89, 25, 97, 101, 103, 107, 109, 12, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 10, 173, 179, 181, 191, 193, 197, 199, 49, 211, 223, 227, 229, 233
Offset: 1
Examples
After a(2)=2, the next term that shares a common factor with 2 is a(7)=4, which is permitted since the difference 7-2 = 5 is greater than 4.
Links
- Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1500 from Michael De Vlieger)
- Michael S. Branicky, Python program
- Russ Cox, Proof that every number appears
- Michael De Vlieger, Annotated log log scatterplot of a(n), n = 1..20000, showing primes in red, proper prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where purple represents powerful numbers that are not prime powers. Composite terms are enlarged for visibility.
Programs
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Maple
N:= 1000: # for terms before the first term > N Cands:= [$2..N]: R:= [1]: x:= 1: for n from 2 do found:= false; for j from 1 to N - n do if andmap(t -> igcd(t, Cands[j]) = 1, [seq(R[n-i],i=1 .. min(Cands[j],n-1))]) then found:= true; x:= Cands[j]; R:= [op(R),x]; Cands:= subsop(j=NULL,Cands); break fi od: if not found then break fi od: R; # Robert Israel, Feb 14 2025 -
Mathematica
nn = 120; c[_] = False; u = v = 2; a[1] = 1; Do[k = u; While[Or[c[k], ! CoprimeQ[k, Product[a[h], {h, n - Min[k, n - 1], n - 1}] ] ], If[k > n - 1, k = v, k++]]; Set[{a[n], c[k]}, {k, True}]; If[k == u, While[c[u], u++]]; If[k == v, While[Or[c[v], CompositeQ[v]], v++]], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Feb 14 2025 *) -
Python
# see link for faster version from math import gcd from itertools import count, islice def agen(): # generator of terms alst, aset, an, m = [1], {1}, 1, 2 for n in count(2): yield an an = next(k for k in count(m) if k not in aset and all(gcd(alst[-j], k) == 1 for j in range(1, min(k, n-1)+1))) alst.append(an) aset.add(an) while m in aset: m += 1 print(list(islice(agen(), 61))) # Michael S. Branicky, Feb 13 2025
Extensions
More terms from Michael S. Branicky, Feb 13 2025
Comments