A080196 - OEIS

%I #13 Oct 23 2024 00:42:24

%S 13,26,39,52,65,78,91,104,117,130,143,156,169,182,195,208,234,260,273,

%T 286,312,325,338,351,364,390,416,429,455,468,507,520,546,572,585,624,

%U 637,650,676,702,715,728,780,819,832,845,858,910,936,975,1001,1014,1040

%N 13-smooth numbers which are not 11-smooth.

%C Numbers of the form 2^r*3^s*5^t*7^u*11^v*13^w with r, s, t, u, v >= 0, w > 0.

%H Amiram Eldar, <a href="/A080196/b080196.txt">Table of n, a(n) for n = 1..10000</a>

%F From _Amiram Eldar_, Nov 10 2020: (Start)

%F a(n) = 13 * A080197(n).

%F Sum_{n>=1} 1/a(n) = 77/192. (End)

%e 78 = 2*3*13 is a term but 77 = 7*11 is not.

%t Select[Range[1000], FactorInteger[#][[-1, 1]] == 13 &] (* _Amiram Eldar_, Nov 10 2020 *)

%o (PARI) {m=1040; 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=0,floor(log(m/d)/log(11)), e=d*11^v; for(w=1,floor(log(m/e)/log(13)),z=concat(z,e*13^w))))))); z=vecsort(z); for(i=1,length(z),print1(z[i],","))}

%o (Python)

%o from sympy import integer_log, prevprime

%o def A080196(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,13)

%o     return 13*bisection(f,n,n) # _Chai Wah Wu_, Oct 22 2024

%Y Cf. A080197, A051038.

%K easy,nonn

%O 1,1

%A _Klaus Brockhaus_, Feb 10 2003