This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A064775 #43 Dec 09 2024 22:03:26
%S A064775 1,1,1,2,2,2,2,3,4,4,4,5,5,5,5,6,6,7,7,7,7,7,7,8,9,9,10,10,10,11,11,
%T A064775 12,12,12,12,13,13,13,13,14,14,14,14,14,15,15,15,16,17,18,18,18,18,19,
%U A064775 19,20,20,20,20,21,21,21,22,23,23,23,23,23,23,24,24,25,25,25,26,26,26,26
%N A064775 Number of positive integers k <= n such that all prime divisors of k are <= sqrt(k).
%C A064775 A048098(n) is the n-th number k such that all prime divisors of k are <= sqrt(k).
%D A064775 D. P. Parent, Exercices de théorie des nombres, Les grands classiques, Gauthier-Villars, Edition Jacques Gabay, p. 17.
%H A064775 Harry J. Smith, <a href="/A064775/b064775.txt">Table of n, a(n) for n = 1..1000</a>
%F A064775 a(n) = n - (Sum_{p<=sqrt(n)} (p-1)) - Sum_{sqrt(n)<p<=n} floor(n/p). a(n) is the largest k such that A048098(k) <= n. Asymptotically: a(n) = (1-log(2))*n + O(n/log(n)).
%F A064775 From _Ridouane Oudra_, Nov 07 2019: (Start)
%F A064775 a(n) = n - Sum_{i=1..floor(sqrt(n))} (pi(floor(n/i)) - pi(i)).
%F A064775 a(n) = n - A242493(n). (End)
%e A064775 Below 28, only k=27,25,24,18,16,12,9,8,4,1 have all their prime divisors less than or equal to sqrt(k), hence a(28)=10. To obtain from A048098(n): A048098(10) = 27 <= 28 < A048098(11)=30, hence a(28)=10.
%o A064775 (PARI) a(n)=n-sum(k=1,sqrtint(n),(k-1)*isprime(k)) - sum(k=sqrtint(n)+1, n, floor(n/k)*isprime(k))
%o A064775 (Magma) [1] cat [#[k:k in [1..n]|forall{p:p in PrimeDivisors(k)| p le Sqrt(k)}]: n in [2..80]]; // _Marius A. Burtea_, Nov 08 2019
%o A064775 (Python)
%o A064775 from math import isqrt
%o A064775 from sympy import primepi
%o A064775 def A064775(n): return int(n+sum(primepi(i)-primepi(n//i) for i in range(1,isqrt(n)+1))) # _Chai Wah Wu_, Oct 05 2024
%Y A064775 Cf. A048098, A242493.
%Y A064775 The following are all different versions of sqrt(n)-smooth numbers: A048098, A063539, A064775, A295084, A333535, A333536.
%K A064775 easy,nonn
%O A064775 1,4
%A A064775 _Benoit Cloitre_, May 11 2002