A252865 a(n) = n if n <= 3, otherwise the smallest squarefree number not occurring earlier having at least one common factor with a(n-2), but none with a(n-1).

1, 2, 3, 10, 21, 5, 6, 35, 22, 7, 11, 14, 33, 26, 15, 13, 30, 91, 34, 39, 17, 42, 85, 38, 51, 19, 66, 95, 46, 55, 23, 65, 69, 70, 57, 58, 93, 29, 31, 87, 62, 105, 74, 77, 37, 110, 111, 82, 129, 41, 43, 123, 86, 141, 106, 47, 53, 94, 159, 118, 165, 59, 78, 295

Offset: 1

Author

Franklin T. Adams-Watters, Dec 23 2014

Keywords

• nonn

Comments

Similar to A098550, but the restriction to squarefree makes it more a sequence of sets of primes, represented by their product.

The sequence has consecutive primes at indices 2 (2 & 3), 10 (7 & 11), 38 (29 & 31), 50 (41 & 43), and 56 (47 & 53). We conjecture that there are no further such pairs.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, arXiv preprint arXiv:1501.01669, 2015 and J. Int. Seq. 18 (2015) 15.6.7.

Crossrefs

Cf. A098550, A005117, A251391, A252867, A252868.

Programs

Haskell

import Data.List (delete)
a252865 n = a252865_list !! (n-1)
a252865_list = 1 : 2 : 3 : f 2 3 (drop 3 a005117_list) where
   f u v ws = g ws where
     g (x:xs) = if gcd x u > 1 && gcd x v == 1
                   then x : f v x (delete x ws) else g xs
-- Reinhard Zumkeller, Dec 24 2014

Mathematica

a[n_ /; n <= 3] = n;
a[n_] := a[n] = For[k = 1, True, k++, If[SquareFreeQ[k], If[FreeQ[Array[a, n-1], k], If[!CoprimeQ[k, a[n-2]] && CoprimeQ[k, a[n-1]], Return[k]]]]];
Array[a, 100] (* Jean-François Alcover, Sep 02 2018 *)

PARI

invecn(v, k, x)=for(i=1, k, if(v[i]==x, return(i))); 0
alist(n)=local(v=vector(n,i,i), x); for(k=4, n, x=4; while(!issquarefree(x)||invecn(v, k-1, x)||gcd(v[k-2], x)==1||gcd(v[k-1],x)!=1, x++); v[k]=x); v

Python

from math import gcd
from sympy import factorint
A252865_list, l1, l2, s, b = [1,2,3], 3, 2, 4, set()
for _ in range(10**2):
    i = s
    while True:
        if max(factorint(i).values()) == 1:
            if not i in b and gcd(i,l1) == 1 and gcd(i,l2) > 1:
                A252865_list.append(i)
                l2, l1 = l1, i
                b.add(i)
                while s in b:
                    b.remove(s)
                    s += 1
                break
        else:
            b.add(i)
        i += 1 # Chai Wah Wu, Dec 24 2014

Python

from math import gcd
from sympy import factorint
from itertools import count, islice
def issquarefree(n): return max(factorint(n).values()) == 1
def agen(): # generator of terms
    aset, an2, an1, m = {1, 2, 3}, 2, 3, 4
    yield from sorted(aset)
    while True:
        an = next(k for k in count(m) if k not in aset and gcd(k, an2) > 1 and gcd(k, an1) == 1 and issquarefree(k))
        an2, an1 = an1, an
        while m in aset or not issquarefree(m): m += 1
        aset.add(an)
        yield an
print(list(islice(agen(), 64))) # Michael S. Branicky, Jan 10 2025