%I #52 Oct 13 2022 13:08:00
%S 1,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,
%T 107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,
%U 197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283
%N Smallest prime factor is >= 17.
%C Also the 17-rough numbers: positive integers that have no prime factors less than 17. - _Michael B. Porter_, Oct 10 2009
%C a(n) - (1001/192) n is periodic with period 5760. - _Robert Israel_, Mar 18 2016
%C From _Peter Bala_, May 12 2018: (Start)
%C The product of two 17-rough numbers is a 17-rough number and the prime factors of a 17-rough number are 17-rough numbers.
%C Let k equal either 13, 14, 15 or 16. Then the product of k numbers n*(n + a)*(n + 2*a)*...*(n + (k-1)*a) in arithmetical progression is divisible by k! for all integer n if and only if a is a 17-rough number.
%C The sequence terms satisfy the congruence x^60 = 1 (mod 30030), where 30030 = 2*3*5*7*11*13. (End)
%C The asymptotic density of this sequence is 192/1001. - _Amiram Eldar_, Sep 30 2020
%H Robert Israel, <a href="/A008366/b008366.txt">Table of n, a(n) for n = 1..10000</a>
%H Peter Bala, <a href="/A008366/a008366_1.pdf">A property of p-rough numbers</a>.
%H Benedict Irwin, <a href="/A008366/a008366.txt">Generating Function</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RoughNumber.html">Rough Number</a>.
%H <a href="/index/Sk#smooth">Index entries for sequences related to smooth numbers</a>
%F Numbers n > 1 such that ((Sum_{k=1..n} k^10) mod n = 0) and ((Sum_{k=1..n} k^12) mod n = 0) (conjecture). - _Gary Detlefs_, Dec 27 2011
%F a(n) = a(n-1) + a(n-5760) - a(n-5761). - _Vaclav Kotesovec_, Mar 18 2016
%F G.f: x*P(x)/(1 - x - x^5760 + x^5761) where P(x) is a polynomial of degree 5760. - _Benedict W. J. Irwin_, Mar 23 2016
%F a(n) = (1001/192)*n + O(1), where the O(1) term is bounded by +/- 19. - _Charles R Greathouse IV_, Oct 13 2022
%p for i from 1 to 500 do if gcd(i,30030) = 1 then print(i); fi; od;
%t Select[ Range[ 300 ], GCD[ #1, 30030 ]==1& ]
%t Join[{1},Select[Range[300],FactorInteger[#][[1,1]]>=17&]] (* _Harvey P. Dale_, Mar 28 2020 *)
%o (PARI) isA008366(n) = gcd(n,30030)==1 \\ _Michael B. Porter_, Oct 10 2009
%Y For k-rough numbers with other values of k, see A000027 A005408 A007310 A007775 A008364 A008365 A008366 A166061 A166063. - _Michael B. Porter_, Oct 10 2009
%Y Cf. A005867.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_