login
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).
1
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, 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, 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
OFFSET
1,7
COMMENTS
Weak conjecture: sequence gives positive values only; strong conjecture: for n large enough a(n) > 2*sqrt(n).
FORMULA
a(n) = Card( x<=n : A006530(x) == 3 mod 4) - Card( x<=n : A006530(x) == 1 mod 4)
a(n) = A071823(n) - A071824(n). - Sean A. Irvine, Aug 16 2024
MAPLE
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:
ListTools:-PartialSums(map(f, [$1..100])); # Robert Israel, Nov 05 2024
PROG
(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)), ", "))
CROSSREFS
Sequence in context: A147952 A091316 A321862 * A115727 A115726 A086413
KEYWORD
easy,nonn,look,changed
AUTHOR
Benoit Cloitre, Jun 07 2002
EXTENSIONS
Missing a(1)=0 inserted by Sean A. Irvine, Aug 16 2024
STATUS
approved