A353237 a(n) = Sum_{d|n} (-1)^(d'), where d' is the arithmetic derivative of d (A003415).

1, 0, 0, 1, 0, -2, 0, 2, 1, -2, 0, 0, 0, -2, 0, 3, 0, -2, 0, 0, 0, -2, 0, 2, 1, -2, 0, 0, 0, -4, 0, 4, 0, -2, 0, 1, 0, -2, 0, 2, 0, -4, 0, 0, 0, -2, 0, 4, 1, -2, 0, 0, 0, -4, 0, 2, 0, -2, 0, 0, 0, -2, 0, 5, 0, -4, 0, 0, 0, -4, 0, 4, 0, -2, 0, 0, 0, -4, 0, 4, 1, -2, 0, 0, 0, -2

Offset

1,6

Comments

From Robert Israel, Jul 23 2023: (Start)

a(n) = 0 if n is odd and squarefree.

a(n) = 1 if n is a square and not divisible by 16.

a(n) < 0 if n > 2 and n == 2 (mod 4).

a(n) = -2 if n = 2*p where p is an odd prime or the square of an odd prime.

(End)

Links

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

Formula

a(n) = 2*A353236(n) - A000005(n).

a(n) = A000005(n) - 2*A353235(n).

Example

a(6) = Sum_{d|6} (-1)^(d') = (-1)^(1') + (-1)^(2') + (-1)^(3') + (-1)^(6') = (-1)^0 + (-1)^1 + (-1)^1 + (-1)^5 = -2.

Maple

ader:= proc(n) option remember;
  local t;
  n*add(t[2]/t[1], t=ifactors(n)[2])
end proc:
f:= proc(n) local d; add ((-1)^ader(d), d = numtheory:-divisors(n)) end proc:
map(f, [$1..100]); # Robert Israel, Jul 23 2023

Mathematica

d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := DivisorSum[n, (-1)^d[#] &]; Array[a, 100] (* Amiram Eldar, May 02 2022 *)

Prog

(PARI) ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
a(n) = sumdiv(n, d, (-1)^ad(d)); \\ Michel Marcus, May 02 2022

Crossrefs

Cf. A000005 (tau), A003415 (n'), A353235, A353236.

Sequence in context: A166348 A294658 A127543 * A280830 A068907 A219762

Adjacent sequences: A353234 A353235 A353236 * A353238 A353239 A353240

Keyword

sign

Author

Wesley Ivan Hurt, May 01 2022

Status

approved