A353235 Number of divisors of n whose arithmetic derivative is odd.

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 2, 1, 1, 4, 1, 3, 2, 3, 1, 3, 1, 3, 2, 3, 1, 6, 1, 1, 2, 3, 2, 4, 1, 3, 2, 3, 1, 6, 1, 3, 3, 3, 1, 3, 1, 4, 2, 3, 1, 6, 2, 3, 2, 3, 1, 6, 1, 3, 3, 1, 2, 6, 1, 3, 2, 6, 1, 4, 1, 3, 3, 3, 2, 6, 1, 3, 2, 3, 1, 6, 2, 3, 2, 3, 1, 9, 2, 3, 2, 3

Offset

1,6

Links

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

Formula

a(n) = Sum_{d|n} ((d') mod 2).

a(n) = tau(n)/2 - (1/2) * Sum_{d|n} (-1)^(d').

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

a(n) = A000005(n)/2 - A353237(n)/2.

From Robert Israel, Jun 05 2023: (Start)

If n = 2^k * m where m is odd and k >= 1, a(n) = a(m) + A000005(m).

If n is odd and squarefree, a(n) = 2^(A001222(n)-1).

If p is an odd prime, a(p^k) = ceil(k/2).

If k and m are odd, a(k*m) = A000005(k)*a(m) + A000005(m)*a(k) - 2*a(m)*a(k).

(End)

Example

a(12) = 3; 12 has 3 divisors whose arithmetic derivatives are odd: 2' = 1, 3' = 1, and 6' = 5.

Maple

aderodd:= proc(n) local t; option remember;
  (n*add(t[2]/t[1], t=ifactors(n)[2]))::odd
end proc:
f:= proc(n) local t;
  nops(select(aderodd, numtheory:-divisors(n)))
end proc:
map(f, [$1..100]); # Robert Israel, Jun 05 2023

Mathematica

d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := DivisorSum[n, 1 &, OddQ[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, ad(d) % 2); \\ Michel Marcus, May 02 2022

Crossrefs

Cf. A000005 (tau), A003415 (n'), A353236, A353237.

Sequence in context: A095345 A342671 A132468 * A243915 A367482 A367095

Adjacent sequences: A353232 A353233 A353234 * A353236 A353237 A353238

Keyword

nonn

Author

Wesley Ivan Hurt, May 01 2022

Status

approved