cp's OEIS Frontend

A008365 13-rough numbers: positive integers that have no prime factors less than 13.

%I A008365 #73 Feb 16 2025 08:32:32
%S A008365 1,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,
%T A008365 103,107,109,113,127,131,137,139,149,151,157,163,167,169,173,179,181,
%U A008365 191,193,197,199,211,221,223,227,229,233,239,241,247,251,257,263,269
%N A008365 13-rough numbers: positive integers that have no prime factors less than 13.
%C A008365 For n > 1, the smallest prime factor of a(n) is >= 13.
%C A008365 Conjecture: Numbers n such that n^24 is congruent to {1,421,631,841} mod 2310. - _Gary Detlefs_, Dec 30 2011
%C A008365 This sequence is exactly the set of positive values of r such that ( Product_{k = 0..10} n + k*r )/11! is an integer for all n. - _Peter Bala_, Nov 14 2015
%C A008365 The asymptotic density of this sequence is 16/77. - _Amiram Eldar_, Sep 30 2020
%H A008365 Reinhard Zumkeller, <a href="/A008365/b008365.txt">Table of n, a(n) for n = 1..10000</a>
%H A008365 Peter Bala, <a href="/A008366/a008366_1.pdf">A property of p-rough numbers</a>.
%H A008365 Benedict W. J. Irwin, <a href="/A008365/a008365.txt">Generating Function</a>.
%H A008365 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RoughNumber.html">Rough Number</a>.
%H A008365 <a href="/index/Rec#order_481">Index entries for linear recurrences with constant coefficients</a>, order 481.
%H A008365 <a href="/index/Sk#smooth">Index entries for sequences related to smooth numbers</a>
%F A008365 G.f: x*P(x)/(1 - x - x^480 + x^481) where P(x) is a polynomial of degree 480. - _Benedict W. J. Irwin_, Mar 18 2016
%F A008365 77*n/16 - 13 < a(n) < 77*n/16 + 8. - _Charles R Greathouse IV_, Mar 21 2023
%F A008365 a(n) = a(n-1) + a(n-480) - a(n-481). - _Charles R Greathouse IV_, Mar 21 2023
%p A008365 for i from 1 to 500 do if gcd(i,2310) = 1 then print(i); fi; od;
%t A008365 Select[ Range[ 300 ], GCD[ #1, 2310 ]==1& ]
%o A008365 (PARI) isA008365(n) = gcd(n,2310)==1 \\ _Michael B. Porter_, Oct 10 2009
%o A008365 (Haskell)
%o A008365 a008365 n = a008365_list !! (n-1)
%o A008365 a008365_list = 1 : filter ((> 11) . a020639) [1..]
%o A008365 -- _Reinhard Zumkeller_, Jan 06 2013
%Y A008365 For k-rough numbers with other values of k, see A000027, A005408, A007310, A007775, A008364, A008365, A008366, A166061, A166063.
%K A008365 nonn,easy
%O A008365 1,2
%A A008365 _N. J. A. Sloane_
%E A008365 New name following a comment of _Michael B. Porter_, Mar 21 2023