A347870 a(n) = A003415(sigma(n)) mod 2, where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0

Offset

1

Comments

If a(k) = 0 for all terms k of A342923, then there cannot be any odd perfect numbers, as k + 3*A003415(k) is odd for any k of the form 4u+2. See comments in A005820 and A235991, also in A347887.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for characteristic functions

Index entries for sequences related to sigma(n)

Formula

a(n) = A000035(A342925(n)) = A165560(A000203(n)).

a(n) = A000035(n) XOR A347871(n).

Mathematica

ad[1] = 0; ad[n_] := n * Total@(Last[#]/First[#]& /@ FactorInteger[n]); a[n_] := Mod[ad[DivisorSigma[1, n]], 2]; Array[a, 105] (* Amiram Eldar, Sep 18 2021 *)

Prog

(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342925(n) = A003415(sigma(n));
A347870(n) = (A342925(n)%2);

Crossrefs

Cf. A000035, A000203, A003415, A005820, A165560, A235991, A342923, A342925, A347871, A347887, A349909 (partial sums).

Characteristic function of A347877, while its complement A347878 gives the positions of zeros.

Sequence in context: A288741 A341684 A327183 * A188967 A090171 A316832

Adjacent sequences: A347867 A347868 A347869 * A347871 A347872 A347873

Keyword

nonn

Author

Antti Karttunen, Sep 17 2021

Status

approved