A353628 Parity of A001001(n), where A001001(n) = Sum_{d|n} d*sigma(d).

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

Offset

1

Comments

See discussion at A209635.

Links

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

Index entries for characteristic functions

Formula

a(n) = A000035(A001001(n)).

a(n) = a(2n) = a(A000265(n)).

a(n) >= A353627(n).

Multiplicative with a(p^e) = A000035(((p^(e+1)-1)(p^(e+2)-1)) / ((p-1)(p^2-1))). - Antti Karttunen, Dec 20 2022

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi^2/12 = 0.822467... (A072691). - Amiram Eldar, Oct 23 2023

Mathematica

f[p_, e_] := Mod[Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 2}], 2]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 23 2023 *)

Prog

(PARI)
A001001(n) = sumdivmult(n, d, sigma(d)*d); \\ From A001001
A353628(n) = (A001001(n)%2);
(PARI) A353628(n) = { my(f = factor(n)); prod(k=1, #f~, my(p=f[k, 1], e=f[k, 2]); (((p^(e+1)-1)*(p^(e+2)-1)) / ((p-1)*(p^2-1)))%2); }; \\ Antti Karttunen, Dec 20 2022

Crossrefs

Characteristic function of A353511.

Differs from A353627 at the positions given by A353456.

Cf. A000035, A000265, A001001 (inverse Möbius transform of A064987), A072691, A209635.

Sequence in context: A209635 A355689 A353627 * A359551 A353458 A110242

Adjacent sequences: A353625 A353626 A353627 * A353629 A353630 A353631

Keyword

nonn,mult

Author

Antti Karttunen, May 03 2022

Extensions

Keyword:mult added by Antti Karttunen, Dec 20 2022

Status

approved