A080196 13-smooth numbers which are not 11-smooth.

13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156, 169, 182, 195, 208, 234, 260, 273, 286, 312, 325, 338, 351, 364, 390, 416, 429, 455, 468, 507, 520, 546, 572, 585, 624, 637, 650, 676, 702, 715, 728, 780, 819, 832, 845, 858, 910, 936, 975, 1001, 1014, 1040

Offset: 1

Author

Klaus Brockhaus, Feb 10 2003

Keywords

• easy
• nonn

Comments

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.

Examples

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A080197, A051038.

Programs

Mathematica

Select[Range[1000], FactorInteger[#][[-1, 1]] == 13 &] (* Amiram Eldar, Nov 10 2020 *)

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],","))}

Python

from sympy import integer_log, prevprime
def A080196(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,13)
    return 13*bisection(f,n,n) # Chai Wah Wu, Oct 22 2024

Formula

From Amiram Eldar, Nov 10 2020: (Start)

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

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