A252868 Squarefree version of A252867.

1, 2, 3, 10, 21, 5, 6, 35, 22, 15, 14, 33, 7, 30, 77, 26, 105, 13, 42, 65, 66, 91, 11, 70, 143, 210, 187, 39, 55, 78, 385, 34, 165, 182, 51, 110, 273, 85, 154, 195, 119, 330, 17, 231, 170, 429, 238, 715, 102, 455, 374, 1365, 38, 1155, 442, 57, 770, 663, 95, 462, 1105, 114, 1001, 255

Offset

1,2

Comments

a(n) = n if n <= 3, otherwise the first squarefree number not occurring earlier having at least one common factor with a(n-2), but none with a(n-1). The squarefree numbers are ordered by their occurrence in A019565.

These represent the same sets of integers as A252867 does, but using the factorization of squarefree numbers for the representation.

This is a permutation of the squarefree numbers. [I believe this is at present only a conjecture. - N. J. A. Sloane, Jan 10 2015]

Links

Chai Wah Wu, 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.

Formula

a(n)=A019565(A252867(n-1))

Mathematica

(* b = A019565, c = A252867 *)
b[n_] := Times @@ Prime[Flatten[Position[#, 1]]]&[Reverse[IntegerDigits[n, 2]]];
c[n_] := c[n] = If[n<3, n, For[k=3, True, k++, If[FreeQ[Array[c, n-1], k], If[BitAnd[k, c[n-2]] >= 1 && BitAnd[k, c[n-1]] == 0, Return[k]]]]];
a[n_] := b[c[n-1]];
Array[a, 64] (* Jean-François Alcover, Oct 03 2018 *)

Prog

(PARI) invecn(v, k, x)=for(i=1, k, if(v[i]==x, return(i))); 0
squarefree(n)=local(r=1, i=1); while(n>0, if(n%2, r*=prime(i)); i++; n\=2); r
alist(n)=local(v=vector(n, i, i-1), x); for(k=4, n, x=3; while(invecn(v, k-1, x)||!bitand(v[k-2], x)||bitand(v[k-1], x), x++); v[k]=x); vector(n, i, squarefree(v[i]))
(Python)
from operator import mul
from functools import reduce
from sympy import prime
def A019565(n):
....return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1
A252868_list, l1, l2, s, b = [1, 2, 3], 2, 1, 3, set()
for _ in range(10**4):
....i = s
....while True:
........if not (i in b or i & l1) and i & l2:
............A252868_list.append(A019565(i))
............l2, l1 = l1, i
............b.add(i)
............while s in b:
................b.remove(s)
................s += 1
............break
........i += 1 # Chai Wah Wu, Dec 25 2014

Crossrefs

Cf. A252867, A098550, A252865, A048793, A019565.

Sequence in context: A141050 A354720 A252865 * A352394 A225477 A079161

Adjacent sequences: A252865 A252866 A252867 * A252869 A252870 A252871

Keyword

nonn

Author

Franklin T. Adams-Watters, Dec 23 2014

Status

approved