

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
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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 ksmooth (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 711 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



