A353628 - OEIS

%I #21 Oct 23 2023 02:02:08

%S 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,

%T 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,

%U 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

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

%C See discussion at A209635.

%H Antti Karttunen, <a href="/A353628/b353628.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>

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

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

%F a(n) >= A353627(n).

%F Multiplicative with a(p^e) = A000035(((p^(e+1)-1)(p^(e+2)-1)) / ((p-1)(p^2-1))). - _Antti Karttunen_, Dec 20 2022

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

%t 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 *)

%o (PARI)

%o A001001(n) = sumdivmult(n, d, sigma(d)*d); \\ From A001001

%o A353628(n) = (A001001(n)%2);

%o (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

%Y Characteristic function of A353511.

%Y Differs from A353627 at the positions given by A353456.

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

%K nonn,mult

%O 1

%A _Antti Karttunen_, May 03 2022

%E Keyword:mult added by _Antti Karttunen_, Dec 20 2022