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A349136 Möbius transform of Kimberling's paraphrases, A003602. 19
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
LINKS
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. A000004, A072451 (even and odd bisection).
Cf. also A347233, A349127, A349137.
Sequence in context: A035442 A213177 A265017 * A035376 A259708 A029220
KEYWORD
nonn
AUTHOR
Antti Karttunen, Nov 13 2021
STATUS
approved

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)