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

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, 10, 11, 11, 11, 12, 12, 13, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 18, 19, 20, 20, 20, 20, 20, 21, 21, 22, 23, 23, 23, 23, 24, 24, 24, 25, 25, 25, 25, 25, 26, 27, 28, 28, 28, 29, 29

Offset

1,10

Comments

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) )

Partial sums of the indicator function of A071821. - Robert Israel, Nov 05 2024

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Maple

filter:= n -> max(numtheory:-factorset(n)) mod 4 = 1:
R:= NULL: t:= 0:
for i from 1 to 100 do
  if filter(i) then t:= t+1 fi;
  R:= R, t
od:
R; # Robert Israel, Nov 05 2024

Mathematica

Join[{0}, Accumulate[Boole[Divisible[FactorInteger[Range[2, 100]][[All, -1, 1]] - 1, 4]]]] (* Paolo Xausa, Nov 23 2024 *)

Prog

(PARI) a(n)=sum(i=2, n, ((factor(i)[omega(i), 1])-1)%4==0)
(Python)
from sympy import factorint
a = lambda n: sum(1 for i in range(2, n + 2) if (max(factorint(i).keys()) - 1) & 3 == 0)
print([a(n) for n in range(0, 79)]) # Darío Clavijo, Nov 05 2024

Crossrefs

Cf. A071821.

Sequence in context: A078451 A332247 A341168 * A224921 A225370 A114540

Adjacent sequences: A071821 A071822 A071823 * A071825 A071826 A071827

Keyword

easy,nonn

Author

Benoit Cloitre, Jun 07 2002

Extensions

Missing a(1)=0 inserted by Sean A. Irvine, Aug 15 2024

Name edited by Michel Marcus, Nov 05 2024

Status

approved