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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. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

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.

EXAMPLE

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.

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 *)

CROSSREFS

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

Sequence in context: A077178 A267376 A267380 * A322200 A028292 A256244

Adjacent sequences:  A217284 A217285 A217286 * A217288 A217289 A217290

KEYWORD

nonn

AUTHOR

Lior Manor, Sep 30 2012

EXTENSIONS

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

STATUS

approved

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Last modified July 5 20:10 EDT 2020. Contains 335473 sequences. (Running on oeis4.)