A133581 (k^2)-th k-smooth number for k = prime(n).

8, 16, 54, 112, 396, 512, 1008, 1155, 1794, 3312, 3520, 5488, 6776, 7020, 8405, 11180, 14384, 14720, 18241, 20339, 20709, 24769, 27094, 31648, 38994, 41890, 42336, 45318, 45825, 48852, 66234, 69874, 76857, 77441, 91719, 92323, 100215, 108376, 112896, 121539

Offset

1,1

Comments

An integer is k-smooth if it has no prime factors > k.

Links

Table of n, a(n) for n=1..40.

Eric Weisstein's World of Mathematics, Smooth Number

Formula

a(n) = A001248(n)-th integer which has no prime factors > A000040(n).

Example

a(1) = 8 = A000079(4).
a(2) = 16 = A003586(9).
a(3) = 54 = A051037(25).

Prog

(Python)
from sympy import integer_log, prime, prevprime
def A133581(n):
    if n==1: return 8
    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))
    k = prime(n)
    def f(x): return k**2+x-g(x, k)
    return bisection(f, k**2, k**2) # Chai Wah Wu, Sep 17 2024

Crossrefs

Cf. A000040, A000079, A001248, A003586, A051037, A002473, A051038.

Sequence in context: A235055 A209997 A212764 * A222544 A307876 A166638

Adjacent sequences: A133578 A133579 A133580 * A133582 A133583 A133584

Keyword

nonn,less

Author

Jonathan Vos Post, Dec 26 2007

Extensions

Corrected and extended by D. S. McNeil, Dec 08 2010

a(33)-a(40) from Chai Wah Wu, Sep 17 2024

Status

approved