A329393 - OEIS

%I #23 Nov 18 2024 19:02:39

%S -1,0,-2,3,0,-2,0,4,-3,0,-2,4,0,-4,3,0,4,-2,0,-4,4,0,4,-3,0,4,-4,0,-2,

%T 6,0,-6,3,0,4,-2,0,4,-4,4,0,-4,0,6,-5,4,0,-2,4,0,-6,0,6,-2,4,0,-6,5,0,

%U -4,4,0,4,-4,0,4,-2,4,0,4,-8,0,6

%N Number of odd divisors minus number of even divisors of the n-th composite.

%C The mode, or most frequent value of this sequence is 0, which corresponds to composites with equal number of odd and even divisors, A016825(n), n >= 1. The next most frequent value is 4.

%C The value 2 does not appear in this sequence, in contrast to A048272, where A048272(p)=2 for every p = odd prime.

%H Harvey P. Dale, <a href="/A329393/b329393.txt">Table of n, a(n) for n = 1..3000</a>

%e a(1) = -1 since the first composite number is 4, which has 1 odd divisor (1), and 2 even divisors (2,4).

%e a(2) = 0 since the second composite number is 6, which has 2 odd divisors (1,3) and 2 even divisors (2,6).

%t f[p_, e_] := If[p == 2, 1 - e, 1 + e]; diffNum[1] = 1; diffNum[n_] := Times @@ (f @@@ FactorInteger[n]); diffNum /@ Select[Range[100], CompositeQ] (* _Amiram Eldar_, Nov 25 2019 *)

%t odmed[n_]:=With[{divs=Divisors[n]},Count[divs,_?OddQ]-Count[divs,_?EvenQ]]; odmed/@Select[Range[100],CompositeQ] (* _Harvey P. Dale_, Nov 18 2024 *)

%Y Cf. A002808, A016825, A048272.

%K sign

%O 1,3

%A _Enrique Navarrete_, Nov 12 2019