cp's OEIS Frontend

A071824 Number of integers <= n whose largest prime factor is of the form 4*k+1.

%I A071824 #31 Nov 24 2024 02:08:20
%S A071824 0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,5,5,5,6,6,6,6,6,7,8,8,8,9,10,10,10,
%T A071824 10,11,11,11,12,12,13,14,15,15,15,15,16,16,16,16,16,17,18,19,20,20,20,
%U A071824 20,20,21,21,22,23,23,23,23,24,24,24,25,25,25,25,25,26,27,28,28,28,29,29
%N A071824 Number of integers <= n whose largest prime factor is of the form 4*k+1.
%C A071824 Conjecture : for n > 1000 a(n) - n/2 < - sqrt(n) ( if b(n) denotes the number of x with largest prime factor of the form 4k+3 less than or equal to n, it is conjectured that if n > 1000 b(n)- n/2 > sqrt(n) )
%C A071824 Partial sums of the indicator function of A071821. - _Robert Israel_, Nov 05 2024
%H A071824 Robert Israel, <a href="/A071824/b071824.txt">Table of n, a(n) for n = 1..10000</a>
%p A071824 filter:= n -> max(numtheory:-factorset(n)) mod 4 = 1:
%p A071824 R:= NULL: t:= 0:
%p A071824 for i from 1 to 100 do
%p A071824   if filter(i) then t:= t+1 fi;
%p A071824   R:= R,t
%p A071824 od:
%p A071824 R; # _Robert Israel_, Nov 05 2024
%t A071824 Join[{0}, Accumulate[Boole[Divisible[FactorInteger[Range[2, 100]][[All, -1, 1]] - 1, 4]]]] (* _Paolo Xausa_, Nov 23 2024 *)
%o A071824 (PARI) a(n)=sum(i=2, n, ((factor(i)[omega(i),1])-1)%4==0)
%o A071824 (Python)
%o A071824 from sympy import factorint
%o A071824 a = lambda n: sum(1 for i in range(2, n + 2) if (max(factorint(i).keys()) - 1) & 3 == 0)
%o A071824 print([a(n) for n in range(0, 79)]) # _Darío Clavijo_, Nov 05 2024
%Y A071824 Cf. A071821.
%K A071824 easy,nonn
%O A071824 1,10
%A A071824 _Benoit Cloitre_, Jun 07 2002
%E A071824 Missing a(1)=0 inserted by _Sean A. Irvine_, Aug 15 2024
%E A071824 Name edited by _Michel Marcus_, Nov 05 2024