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

1,1

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.

Links

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

Formula

From Amiram Eldar, Nov 10 2020: (Start)

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

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A080197, A051038.

Sequence in context: A044898 A048842 A008595 * A033025 A044838 A033010

Adjacent sequences: A080193 A080194 A080195 * A080197 A080198 A080199

Keyword

easy,nonn

Author

Klaus Brockhaus, Feb 10 2003

Status

approved