A217287 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.

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, 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, 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

Offset: 1

Author

Lior Manor, Sep 30 2012

Keywords

• nonn

Comments

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.

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

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Lior Manor, First 1000 entries with the associated chains (For n=1, the chain 1,2,3 should be added. - _N. J. A. Sloane_, Apr 25 2020)
• J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000 [math.NT], April 2020.

Crossrefs

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

Programs

Maple

A006530 := n->max(1, op(numtheory[factorset](n)));
a:=[]; M:=120;
for n from 1 to M do
   for k from 1 to 3*n do
   if A006530(n+k) <= k then a:=[op(a),k]; break; fi;
   od;
od:
a; # N. J. A. Sloane, Apr 25 2020

Mathematica

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 *)

Extensions

a(1) = 3 added by N. J. A. Sloane, Apr 25 2020