A107352 Number of positive integers <= 10^n that are divisible by no prime exceeding 11.

1, 10, 55, 192, 522, 1197, 2432, 4520, 7838, 12867, 20193, 30524, 44696, 63694, 88658, 120895, 161885, 213294, 276997, 355082, 449849, 563834, 699826, 860861, 1050260, 1271598, 1528765, 1825937, 2167611, 2558606, 3004075, 3509523

Offset: 0

Author

N. J. A. Sloane, May 23 2005

Keywords

• nonn

Comments

Lehmer quotes A. E. Western as computing a(5) = 1197, a(8) = 7838 and a(10) = 20193.

Number of integers of the form 2^a*3^b*5^c*7^d*11^e <= 10^n.

Links

• David A. Corneth, Table of n, a(n) for n = 0..119
• D. H. Lehmer, The lattice points of an n-dimensional tetrahedron, Duke Math. J., 7 (1941), 341-353.

Crossrefs

Row 5 of A253635.

Cf. A011557, A051038.

Programs

Mathematica

fQ[n_] := FactorInteger[n][[ -1, 1]] < 13; c = 1; k = 1; Do[ While[k <= 10^n, If[ fQ[k], c++ ]; k++ ]; Print[c], {n, 0, 9}] (* Or *)
n = 32; t = Select[ Flatten[ Table[11^e*Select[ Flatten[ Table[7^d*Select[ Flatten[ Table[5^c*Select[ Flatten[ Table[2^a*3^b, {a, 0, Log[2, 10^n]}, {b, 0, Log[3, 10^n]}]], # <= 10^n &], {c, 0, Log[5, 10^n]}]], # <= 10^n &], {d, 0, Log[7, 10^n]}]], # <= 10^n &], {e, 0, Log[11, 10^n]}]], # <= 10^n &]; Table[ Length[ Select[t, # <= 10^n &]], {n, 0, 32}] (* Robert G. Wilson v, May 24 2005 *)

Python

from sympy import integer_log, prevprime
def A107352(n):
    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))
    return g(10**n,11) # Chai Wah Wu, Oct 22 2024

Formula

Does a(n)/(a(n-1) - a(n-2)) tend to c*n + d for large n where c ~= 0.20 and d ~= 1.37? - David A. Corneth, Nov 14 2019

Extensions

More terms from Robert G. Wilson v and Don Reble, May 26 2005