A349136 Möbius transform of Kimberling's paraphrases, A003602.

1, 0, 1, 0, 2, 0, 3, 0, 3, 0, 5, 0, 6, 0, 4, 0, 8, 0, 9, 0, 6, 0, 11, 0, 10, 0, 9, 0, 14, 0, 15, 0, 10, 0, 12, 0, 18, 0, 12, 0, 20, 0, 21, 0, 12, 0, 23, 0, 21, 0, 16, 0, 26, 0, 20, 0, 18, 0, 29, 0, 30, 0, 18, 0, 24, 0, 33, 0, 22, 0, 35, 0, 36, 0, 20, 0, 30, 0, 39, 0, 27, 0, 41, 0, 32, 0, 28, 0, 44, 0, 36, 0, 30, 0, 36

Offset

1,5

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

a(n) = Sum_{d|n} A008683(d) * A003602(n/d).

a(1) = 1, a(n) = A000010(n)/2 for odd n > 1, a(n) = 0 for even n.

For all n >= 1, a(2*n-1) = A055034(2*n-1) = A072451(n).

a(n) = phi(n) - (1/2)*phi(2n), for n>1. - Ridouane Oudra, Jul 13 2023

Sum_{k=1..n} a(k) ~ (1/Pi^2)*n^2. - Amiram Eldar, Jul 15 2023

Maple

with(numtheory): a:=proc(n) if n=1 then 1; elif n mod 2 = 0 then 0; else phi(n)/2; fi: end proc: seq(a(n), n=1..60); # Ridouane Oudra, Jul 13 2023

Mathematica

k[n_] := (n/2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, MoebiusMu[#] * k[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 13 2021 *)

Prog

(PARI) A349136(n) = if(1==n, 1, if(n%2, eulerphi(n)/2, 0));
(PARI)
A003602(n) = (1+(n>>valuation(n, 2)))/2;
A349136(n) = sumdiv(n, d, moebius(d)*A003602(n/d));
(Python)
from sympy import totient
def A349136(n): return totient(n)+1>>1 if n&1 else 0 # Chai Wah Wu, Nov 24 2023

Crossrefs

Agrees with A055034 on odd arguments.

Cf. A000010, A003602, A008683, A349134.

Cf. A000004, A072451 (even and odd bisection).

Cf. also A347233, A349127, A349137.

Sequence in context: A035442 A213177 A265017 * A035376 A259708 A029220

Adjacent sequences: A349133 A349134 A349135 * A349137 A349138 A349139

Keyword

nonn

Author

Antti Karttunen, Nov 13 2021

Status

approved