%I #19 Oct 23 2024 00:42:14

%S 11,22,33,44,55,66,77,88,99,110,121,132,154,165,176,198,220,231,242,

%T 264,275,297,308,330,352,363,385,396,440,462,484,495,528,539,550,594,

%U 605,616,660,693,704,726,770,792,825,847,880,891,924,968,990,1056,1078,1089

%N 11-smooth numbers which are not 7-smooth.

%C Numbers of the form 2^r*3^s*5^t*7^u*11^v with r, s, t, u >= 0, v > 0.

%H Robert Israel, <a href="/A080195/b080195.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = 11 * A051038(n). - _David A. Corneth_, May 27 2017

%F Sum_{n>=1} 1/a(n) = 7/16. - _Amiram Eldar_, Nov 10 2020

%e 33 = 3*11 is a term but 35 = 5*7 is not.

%p N:= 10^6; # to get all terms <= N

%p A:= NULL;

%p for v from 1 to floor(log[11](N)) do

%p V:= 11^v;

%p for u from 0 to floor(log[7](N/V)) do

%p U:= 7^u*V;

%p for t from 0 to floor(log[5](N/U)) do

%p T:= 5^t*U;

%p for s from 0 to floor(log[3](N/T)) do

%p S:= 3^s*T;

%p for r from 0 to floor(log[2](N/S)) do

%p A:= A, 2^r*S

%p od

%p od

%p od

%p od

%p od:

%p {A}; # _Robert Israel_, May 28 2014

%t Select[Range[1000], FactorInteger[#][[-1, 1]] == 11 &] (* _Amiram Eldar_, Nov 10 2020 *)

%o (PARI) {m=1100; z=[]; for(r=0,floor(log(m)/log(2)),a=2^r; for(s=0,floor(log(m/a)/log(3)),b=a*3^s; for(t=0, floor(log(m/b)/log(5)),c=b*5^t; for(u=0,floor(log(m/c)/log(7)),d=c*7^u; for(v=1,floor(log(m/d)/log(11)), z=concat(z,d*11^v)))))); z=vecsort(z); for(i=1,length(z),print1(z[i],","))}

%o (Python)

%o from sympy import integer_log, prevprime

%o def A080195(n):

%o def bisection(f,kmin=0,kmax=1):

%o while f(kmax) > kmax: kmax <<= 1

%o while kmax-kmin > 1:

%o kmid = kmax+kmin>>1

%o if f(kmid) <= kmid:

%o kmax = kmid

%o else:

%o kmin = kmid

%o return kmax

%o def g(x,m): return sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1)) if m==3 else sum(g(x//(m**i),prevprime(m))for i in range(integer_log(x,m)[0]+1))

%o def f(x): return n+x-g(x,11)

%o return 11*bisection(f,n,n) # _Chai Wah Wu_, Oct 22 2024

%Y Cf. A051038, A002473.

%K easy,nonn

%O 1,1

%A _Klaus Brockhaus_, Feb 10 2003