|
|
A058989
|
|
Largest number of consecutive integers such that each is divisible by a prime <= the n-th prime.
|
|
6
|
|
|
1, 3, 5, 9, 13, 21, 25, 33, 39, 45, 57, 65, 73, 89, 99, 105, 117, 131, 151, 173, 189, 199, 215, 233, 257, 263, 281, 299, 311, 329, 353, 377, 387, 413, 431, 449, 475, 491, 509, 537, 549, 573, 599, 615, 641, 659, 685, 717, 741, 761, 797, 809, 833, 857, 875, 907, 925, 953, 977, 1001, 1029, 1057, 1097, 1109
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Marty Weissman conjectured that a(n)=2q-1, where q is the largest prime smaller than the n-th prime. The conjecture holds for the first few terms, but then a(n) is larger than 2q-1. Phil Carmody proved a(n)>=2q-1. Terms were calculated by Weissman, Carmody and McCranie.
|
|
REFERENCES
|
Dickson, L. E., History of the Theory of Numbers, Vol. 1, p. 439, Chelsea, 1952.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = A048670(n) - 1. See that entry for additional information.
|
|
EXAMPLE
|
The 4th prime is 7. Nine is the maximum number of consecutive integers such that each is divisible by 2, 3, 5 or 7. (Example: 2 through 10) So a(4)=9.
|
|
MATHEMATICA
|
(* This program is not suitable to compute more than a few terms *)
primorial[n_] := Product[Prime[k], {k, 1, n}];
j[n_] := Module[{L = 1, m = 1}, For[k = 2, k <= n+1, k++, If[GCD[k, n] == 1, If[L+m < k, m = k-L]; L = k]]; m];
a[1] = 1;
a[n_] := a[n] = j[primorial[n]] - 1;
|
|
CROSSREFS
|
|
|
KEYWORD
|
nice,nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
a(46) corrected and a(50)-a(54) added by Mario Ziller, Dec 08 2016
|
|
STATUS
|
approved
|
|
|
|