cp's OEIS Frontend

A243285 Number of integers 1 <= k <= n which are not divisible by the square of their largest noncomposite divisor.

%I A243285 #19 Oct 14 2023 15:43:31
%S A243285 0,1,2,2,3,4,5,5,5,6,7,8,9,10,11,11,12,12,13,14,15,16,17,18,18,19,19,
%T A243285 20,21,22,23,23,24,25,26,26,27,28,29,30,31,32,33,34,35,36,37,38,38,38,
%U A243285 39,40,41,41,42,43,44,45,46,47,48,49,50,50,51,52,53,54,55,56,57
%N A243285 Number of integers 1 <= k <= n which are not divisible by the square of their largest noncomposite divisor.
%C A243285 a(n) tells how many natural numbers <= n there are which are not divisible by the square of their largest noncomposite divisor.
%C A243285 The largest noncomposite divisor of 1 is 1 itself, and 1 is divisible by 1^2, thus 1 is not included in the count, and a(1)=0.
%C A243285 The "largest noncomposite divisor" for any integer > 1 means the same thing as the largest prime divisor, and thus we are counting the terms of A102750 (Numbers n such that square of largest prime dividing n does not divide n).
%C A243285 Thus this is the partial sums of the characteric function for A102750.
%H A243285 Antti Karttunen, <a href="/A243285/b243285.txt">Table of n, a(n) for n = 1..10000</a>
%F A243285 a(n) = n - A243283(n).
%F A243285 For all n, a(A102750(n)) = n, thus this sequence works also as an inverse function for the injection A102750.
%e A243285 For n = 9, there are numbers 2, 3, 5, 6 and 7 which are not divisible by the square of their largest prime factor, while 1 is excluded (no prime factors) and 4 and 8 are divisible both by 2^2 and 9 is divisible by 3^2. Thus a(9) = 5.
%t A243285 ndsQ[n_]:=Mod[n,Max[Select[Divisors[n],!CompositeQ[#]&]]^2]!=0; Accumulate[Table[If[ ndsQ[n],1,0],{n,80}]] (* _Harvey P. Dale_, Oct 14 2023 *)
%o A243285 (Scheme) (define (A243285 n) (- n (A243283 n)))
%o A243285 (Python)
%o A243285 from sympy import primefactors
%o A243285 def a243285(n): return 0 if n==1 else sum([1 for k in range(2, n + 1) if k%(primefactors(k)[-1]**2)!=0]) # _Indranil Ghosh_, Jun 15 2017
%Y A243285 Cf. A243282, A243283, A102750, A070003, A013928, A057627.
%K A243285 nonn
%O A243285 1,3
%A A243285 _Antti Karttunen_, Jun 02 2014