A251756 a(0) = 0; for n>0, a(n) is the smallest integer not already in the list with a composite common factor with a(n-1).

0, 4, 8, 12, 6, 18, 9, 27, 36, 16, 20, 10, 30, 15, 45, 54, 24, 28, 14, 42, 21, 63, 72, 32, 40, 44, 22, 66, 33, 99, 81, 90, 48, 52, 26, 78, 39, 117, 108, 56, 60, 50, 25, 75, 100, 64, 68, 34, 102, 51, 153, 126, 70, 35, 105, 84, 76, 38, 114, 57, 171, 135, 120, 80

Offset: 0

Author

Franklin T. Adams-Watters, Dec 08 2014

Keywords

• nonn

Comments

It appears that this sequence includes every composite number.

The values are grouped close to five lines extending from the origin with respective slope of approximately { 0.608, 0.912, 1.22, 1.82, 2.74 } = {1, 1.5, 2, 3, 4.5} * 0.608. (As in A098550 these lines are not really straight.) - M. F. Hasler, Dec 14 2014

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..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. A064413, A002808, A010051, A098550.

Programs

Haskell

import Data.List (delete)
a251756 n = a251756_list !! (n-1)
a251756_list = 0 : f 0 a002808_list where
   f x zs = g zs where
     g (y:ys) | d == 1 || a010051' d == 1 = g ys
              | otherwise = y : f y (delete y zs)
              where d = gcd x y
-- Reinhard Zumkeller, Dec 08 2014

Mathematica

g[a_List] := Block[{k = 4}, While[Not[CompositeQ[GCD[a[[-1]], k]]] || MemberQ[a, k], k++]; Append[a, k]]; Nest[g, {0}, 63] (* L. Edson Jeffery, Dec 08 2014 (after Robert G. Wilson v) *)

PARI

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

Python

from gmpy2 import gcd, is_prime
A251756_list, l, s, b = [0], 0, 1, {}
for _ in range(10**3):
    i = s
    while True:
        if not i in b:
            m = gcd(i, l)
            if not (m == 1 or is_prime(m)):
                A251756_list.append(i)
                l, b[i] = i, True
                while s in b:
                    b.pop(s)
                    s += 1
                break
        i += 1 # Chai Wah Wu, Dec 08 2014