cp's OEIS Frontend

A071825 a(n) = (number of x with largest prime factor of the form 4k+3 less than or equal to n) minus (number of x with largest prime factor of the form 4k+1 less than or equal to n).

%I A071825 #15 Nov 05 2024 21:15:02
%S A071825 0,0,1,1,0,1,2,2,3,2,3,4,3,4,3,3,2,3,4,3,4,5,6,7,6,5,6,7,6,5,6,6,7,6,
%T A071825 7,8,7,8,7,6,5,6,7,8,7,8,9,10,11,10,9,8,7,8,9,10,11,10,11,10,9,10,11,
%U A071825 11,10,11,12,11,12,13,14,15,14,13,12,13,14,13,14,13,14,13,14,15,14,15,14,15
%N A071825 a(n) = (number of x with largest prime factor of the form 4k+3 less than or equal to n) minus (number of x with largest prime factor of the form 4k+1 less than or equal to n).
%C A071825 Weak conjecture: sequence gives positive values only; strong conjecture: for n large enough a(n) > 2*sqrt(n).
%H A071825 Robert Israel, <a href="/A071825/b071825.txt">Table of n, a(n) for n = 1..10000</a>
%H A071825 Robert Israel, <a href="/A071825/a071825.png">Log-log plot of (n, a(n)) for 1 <= n <= 10^6</a>
%F A071825 a(n) = Card( x<=n : A006530(x) == 3 mod 4) - Card( x<=n : A006530(x) == 1 mod 4)
%F A071825 a(n) = A071823(n) - A071824(n). - _Sean A. Irvine_, Aug 16 2024
%p A071825 f:= proc(n) local t; t:= max(numtheory:-factorset(n)) mod 4; if t = 3 then 1 elif t= 1 then -1 else 0 fi end proc:
%p A071825 ListTools:-PartialSums(map(f, [$1..100])); # _Robert Israel_, Nov 05 2024
%o A071825 (PARI) for(n=1,200,print1(sum(i=2,n,if((component(component(factor(i),1),omega(i))-3)%4,0,1))-sum(i=2,n,if((component(component(factor(i),1),omega(i))-1)%4,0,1)),","))
%Y A071825 Cf. A071823, A071824.
%K A071825 easy,nonn,look
%O A071825 1,7
%A A071825 _Benoit Cloitre_, Jun 07 2002
%E A071825 Missing a(1)=0 inserted by _Sean A. Irvine_, Aug 16 2024