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

1,2

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

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

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& ]

Prog

(PARI) isA008365(n) = gcd(n, 2310)==1 \\ Michael B. Porter, Oct 10 2009
(Haskell)
a008365 n = a008365_list !! (n-1)
a008365_list = 1 : filter ((> 11) . a020639) [1..]
-- Reinhard Zumkeller, Jan 06 2013

Crossrefs

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

Sequence in context: A075761 A046064 A322274 * A132077 A235154 A045921

Adjacent sequences: A008362 A008363 A008364 * A008366 A008367 A008368

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

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

Status

approved