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

1, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 221, 223, 227, 229, 233, 239, 241, 247, 251, 257, 263, 269

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

For n > 1, the smallest prime factor of a(n) is >= 13.

Conjecture: Numbers n such that n^24 is congruent to {1,421,631,841} mod 2310. - Gary Detlefs, Dec 30 2011

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

The asymptotic density of this sequence is 16/77. - Amiram Eldar, Sep 30 2020

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Peter Bala, A property of p-rough numbers.
• Benedict W. J. Irwin, Generating Function.
• Eric Weisstein's World of Mathematics, Rough Number.
• Index entries for linear recurrences with constant coefficients, order 481.
• Index entries for sequences related to smooth numbers

Crossrefs

For k-rough numbers with other values of k, see A000027, A005408, A007310, A007775, A008364, A008365, A008366, A166061, A166063.

Programs

Haskell

a008365 n = a008365_list !! (n-1)
a008365_list = 1 : filter ((> 11) . a020639) [1..]
-- Reinhard Zumkeller, Jan 06 2013

Maple

for i from 1 to 500 do if gcd(i,2310) = 1 then print(i); fi; od;

Mathematica

Select[ Range[ 300 ], GCD[ #1, 2310 ]==1& ]

PARI

isA008365(n) = gcd(n,2310)==1 \\ Michael B. Porter, Oct 10 2009

Formula

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

77*n/16 - 13 < a(n) < 77*n/16 + 8. - Charles R Greathouse IV, Mar 21 2023

a(n) = a(n-1) + a(n-480) - a(n-481). - Charles R Greathouse IV, Mar 21 2023

Extensions

New name following a comment of Michael B. Porter, Mar 21 2023