A217287 - OEIS

%I #29 May 01 2020 03:18:15

%S 3,2,3,4,3,2,5,4,3,5,5,4,3,2,3,8,7,6,5,4,3,5,4,3,5,6,5,4,3,2,5,4,3,6,

%T 5,9,8,7,6,5,7,6,5,4,3,8,7,6,5,4,3,8,7,6,5,7,7,6,5,4,3,2,7,8,7,6,5,4,

%U 3,5,9,8,7,6,5,5,4,3,11,10,9,8,7,6,5,10,9,8,7,6,5,4,3,6,5,9,8,7,9,8

%N Length of chain of consecutive integers starting with n, where each new integer in the chain has a prime factor which no previous member in the chain has.

%C a(n) >= 2. If n < 2 is prime or prime power, a(n) >= 3. For any n > 1, k > 1, a(n^k - n) <= n.

%C a(n) is also the smallest k>0 such that n+k is k-smooth (i.e. has no prime factor > k). - _N. J. A. Sloane_, Apr 25 2020

%H Alois P. Heinz, <a href="/A217287/b217287.txt">Table of n, a(n) for n = 1..10000</a>

%H Lior Manor, <a href="/A217287/a217287_1.txt">First 1000 entries with the associated chains</a> (For n=1, the chain 1,2,3 should be added. - _N. J. A. Sloane_, Apr 25 2020)

%H J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, <a href="http://arxiv.org/abs/2004.14000">Three Cousins of Recaman's Sequence</a>, arXiv:2004:14000 [math.NT], April 2020.

%e Example: a(7)=5 since 7 starts a chain of 5 integers 7-11 with the following property: 7 is divisible by 7, 8 is divisible by 2, 9 is divisible by 3, 10 is divisible by 5, 11 is divisible by 11. And the next integer 12 is divisible by 2 and 3, both of them are prime factors of prior members in the chain.

%p A006530 := n->max(1, op(numtheory[factorset](n)));

%p a:=[]; M:=120;

%p for n from 1 to M do

%p    for k from 1 to 3*n do

%p    if A006530(n+k) <= k then a:=[op(a),k]; break; fi;

%p    od;

%p od:

%p a; # _N. J. A. Sloane_, Apr 25 2020

%t Block[{nn = 111, r}, r = Prime@ Range[PrimePi@ nn]; r = Table[FromDigits[#, 2] &@ Map[Boole[Mod[n, #] == 0] &, r], {n, nn}]; Array[Block[{k = # + 1, s = r[[#]]}, While[UnsameQ[s, Set[s, BitOr[s, r[[k]] ] ] ], k++]; k - #] &, nn - Ceiling@ Sqrt@ nn] ] (* _Michael De Vlieger_, Apr 30 2020 *)

%Y Cf. A006530, A217288 and A217289 (records), A217438.

%K nonn

%O 1,1

%A _Lior Manor_, Sep 30 2012

%E a(1) = 3 added by _N. J. A. Sloane_, Apr 25 2020