A376740 - OEIS

%I #51 Nov 20 2024 08:38:50

%S 11,13,17,19,22,23,26,29,31,33,34,37,38,39,41,43,44,46,47,51,52,53,55,

%T 57,58,59,61,62,65,66,67,68,69,71,73,74,76,77,78,79,82,83,85,86,87,88,

%U 89,91,92,93,94,95,97,99,102,104,106,110,111,114,115,116,117

%N Numbers that have at least one two-digit prime factor.

%C Subsequence of A068191, first differing at A068191(55) = 101 which is not a term here.

%C Numbers k such that gcd(k,10978895066407230594062391177770267) > 1. - _Chai Wah Wu_, Nov 18 2024 [The big number is A109819(10) - _Alois P. Heinz_, Nov 18 2024]

%C The asymptotic density of this sequence is A051953(A109819(10))/A109819(10) = 1329644281346285477858013527/2807455661493975149742813527 = 0.473611... . - _Amiram Eldar_, Nov 19 2024

%H Kishin Ikemoto, <a href="/A376740/b376740.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>.

%F a(n + A051953(A109819(10))) = a(n) + A109819(10). - _Amiram Eldar_, Nov 19 2024

%e 201 = 3*67 is in this sequence because it has one two-digit prime factor.

%e 202 = 2*101 is not, because neither of them is two-digit.

%p q:= convert(select(isprime, [seq(i,i=11 .. 99, 2)]),`*`):

%p filter:= n -> igcd(n,q) > 1:

%p select(filter, [$1..200]); # _Robert Israel_, Nov 18 2024

%t A376740Q[n_] := AnyTrue[FactorInteger[n][[All, 1]], 7 < # < 101 &];

%t Select[Range[200], A376740Q] (* _Paolo Xausa_, Nov 18 2024 *)

%o (Python)

%o def ok(n): return any(n%p == 0 for p in [11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97])

%o print([k for k in range(1, 118) if ok(k)]) # _Michael S. Branicky_, Oct 15 2024

%o (PARI) is(k) = {forprime(p = 11, 97, if(!(k % p), return(1))); 0;} \\ _Amiram Eldar_, Nov 19 2024

%Y Cf. A051953, A068191, A084891, A109819.

%K nonn,easy,base

%O 1,1

%A _Kishin Ikemoto_, Oct 03 2024