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(k^2)-th k-smooth number for k = prime(n).
1

%I #14 Feb 16 2025 08:33:06

%S 8,16,54,112,396,512,1008,1155,1794,3312,3520,5488,6776,7020,8405,

%T 11180,14384,14720,18241,20339,20709,24769,27094,31648,38994,41890,

%U 42336,45318,45825,48852,66234,69874,76857,77441,91719,92323,100215,108376,112896,121539

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

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

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SmoothNumber.html">Smooth Number</a>

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

%e a(1) = 8 = A000079(4).

%e a(2) = 16 = A003586(9).

%e a(3) = 54 = A051037(25).

%o (Python)

%o from sympy import integer_log, prime, prevprime

%o def A133581(n):

%o if n==1: return 8

%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 k = prime(n)

%o def f(x): return k**2+x-g(x,k)

%o return bisection(f,k**2,k**2) # _Chai Wah Wu_, Sep 17 2024

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

%K nonn,less,changed

%O 1,1

%A _Jonathan Vos Post_, Dec 26 2007

%E Corrected and extended by _D. S. McNeil_, Dec 08 2010

%E a(33)-a(40) from _Chai Wah Wu_, Sep 17 2024