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A080195 11-smooth numbers which are not 7-smooth.

%I A080195 #19 Oct 23 2024 00:42:14
%S A080195 11,22,33,44,55,66,77,88,99,110,121,132,154,165,176,198,220,231,242,
%T A080195 264,275,297,308,330,352,363,385,396,440,462,484,495,528,539,550,594,
%U A080195 605,616,660,693,704,726,770,792,825,847,880,891,924,968,990,1056,1078,1089
%N A080195 11-smooth numbers which are not 7-smooth.
%C A080195 Numbers of the form 2^r*3^s*5^t*7^u*11^v with r, s, t, u >= 0, v > 0.
%H A080195 Robert Israel, <a href="/A080195/b080195.txt">Table of n, a(n) for n = 1..10000</a>
%F A080195 a(n) = 11 * A051038(n). - _David A. Corneth_, May 27 2017
%F A080195 Sum_{n>=1} 1/a(n) = 7/16. - _Amiram Eldar_, Nov 10 2020
%e A080195 33 = 3*11 is a term but 35 = 5*7 is not.
%p A080195 N:= 10^6; # to get all terms <= N
%p A080195 A:= NULL;
%p A080195 for v from 1 to floor(log[11](N)) do
%p A080195    V:= 11^v;
%p A080195    for u from 0 to floor(log[7](N/V)) do
%p A080195       U:= 7^u*V;
%p A080195       for t from 0 to floor(log[5](N/U)) do
%p A080195         T:= 5^t*U;
%p A080195         for s from 0 to floor(log[3](N/T)) do
%p A080195           S:= 3^s*T;
%p A080195           for r from 0 to floor(log[2](N/S)) do
%p A080195              A:= A, 2^r*S
%p A080195           od
%p A080195          od
%p A080195        od
%p A080195     od
%p A080195 od:
%p A080195 {A}; # _Robert Israel_, May 28 2014
%t A080195 Select[Range[1000], FactorInteger[#][[-1, 1]] == 11 &] (* _Amiram Eldar_, Nov 10 2020 *)
%o A080195 (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 A080195 (Python)
%o A080195 from sympy import integer_log, prevprime
%o A080195 def A080195(n):
%o A080195     def bisection(f,kmin=0,kmax=1):
%o A080195         while f(kmax) > kmax: kmax <<= 1
%o A080195         while kmax-kmin > 1:
%o A080195             kmid = kmax+kmin>>1
%o A080195             if f(kmid) <= kmid:
%o A080195                 kmax = kmid
%o A080195             else:
%o A080195                 kmin = kmid
%o A080195         return kmax
%o A080195     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 A080195     def f(x): return n+x-g(x,11)
%o A080195     return 11*bisection(f,n,n) # _Chai Wah Wu_, Oct 22 2024
%Y A080195 Cf. A051038, A002473.
%K A080195 easy,nonn
%O A080195 1,1
%A A080195 _Klaus Brockhaus_, Feb 10 2003