A080195 11-smooth numbers which are not 7-smooth.
11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 154, 165, 176, 198, 220, 231, 242, 264, 275, 297, 308, 330, 352, 363, 385, 396, 440, 462, 484, 495, 528, 539, 550, 594, 605, 616, 660, 693, 704, 726, 770, 792, 825, 847, 880, 891, 924, 968, 990, 1056, 1078, 1089
Offset: 1
Examples
33 = 3*11 is a term but 35 = 5*7 is not.
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
Programs
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Maple
N:= 10^6; # to get all terms <= N A:= NULL; for v from 1 to floor(log[11](N)) do V:= 11^v; for u from 0 to floor(log[7](N/V)) do U:= 7^u*V; for t from 0 to floor(log[5](N/U)) do T:= 5^t*U; for s from 0 to floor(log[3](N/T)) do S:= 3^s*T; for r from 0 to floor(log[2](N/S)) do A:= A, 2^r*S od od od od od: {A}; # Robert Israel, May 28 2014 -
Mathematica
Select[Range[1000], FactorInteger[#][[-1, 1]] == 11 &] (* Amiram Eldar, Nov 10 2020 *)
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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],","))} -
Python
from sympy import integer_log, prevprime def A080195(n): def bisection(f,kmin=0,kmax=1): while f(kmax) > kmax: kmax <<= 1 while kmax-kmin > 1: kmid = kmax+kmin>>1 if f(kmid) <= kmid: kmax = kmid else: kmin = kmid return kmax 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)) def f(x): return n+x-g(x,11) return 11*bisection(f,n,n) # Chai Wah Wu, Oct 22 2024
Formula
a(n) = 11 * A051038(n). - David A. Corneth, May 27 2017
Sum_{n>=1} 1/a(n) = 7/16. - Amiram Eldar, Nov 10 2020
Comments