A349346 -id:A349346 - cp's OEIS frontend

A349344 Dirichlet inverse of A109168, where A109168(n) = (n+A006519(n))/2, and A006519 is the highest power of 2 dividing n.

1, -2, -2, 0, -3, 4, -4, 0, -1, 6, -6, 0, -7, 8, 4, 0, -9, 2, -10, 0, 5, 12, -12, 0, -4, 14, -2, 0, -15, -8, -16, 0, 7, 18, 6, 0, -19, 20, 8, 0, -21, -10, -22, 0, 3, 24, -24, 0, -9, 8, 10, 0, -27, 4, 8, 0, 11, 30, -30, 0, -31, 32, 4, 0, 9, -14, -34, 0, 13, -12, -36, 0, -37, 38, 8, 0, 9, -16, -40, 0, -4, 42, -42, 0

Offset: 1

Antti Karttunen, Nov 15 2021

sign

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

Cf. A109168 (A140472), A349345.

Cf. also A328203, A349134, A349343, A349346, A349353.

up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA109168(n) = ((n+bitand(n, -n))\2); \\ From A109168 by M. F. Hasler, Oct 19 2019 (Cf. A140472).
v349344 = DirInverseCorrect(vector(up_to,n,A109168(n)));
A349344(n) = v349344[n];

a(1) = 1; a(n) = -Sum_{d|n, d < n} A109168(n/d) * a(d).

a(n) = A349345(n) - A109168(n).

A349341 Dirichlet inverse of A026741, which is defined as n if n is odd, n/2 if n is even.

Original entry on oeis.org

1, -1, -3, -1, -5, 3, -7, -1, 0, 5, -11, 3, -13, 7, 15, -1, -17, 0, -19, 5, 21, 11, -23, 3, 0, 13, 0, 7, -29, -15, -31, -1, 33, 17, 35, 0, -37, 19, 39, 5, -41, -21, -43, 11, 0, 23, -47, 3, 0, 0, 51, 13, -53, 0, 55, 7, 57, 29, -59, -15, -61, 31, 0, -1, 65, -33, -67, 17, 69, -35, -71, 0, -73, 37, 0, 19, 77, -39, -79

Offset: 1

Antti Karttunen, Nov 15 2021

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

Agrees with A349343 on odd numbers.
Cf. A026741, A349342.
Cf. also A328203, A349134, A349344, A349346, A349353.

a[1]=1;a[n_]:=-DivisorSum[n,If[OddQ[n/#],n/#,n/(2#)]*a@#&,#Giorgos Kalogeropoulos, Nov 15 2021 *)
f[p_, e_] := If[e == 1, -p, 0]; f[2, e_] := -1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2023 *)

A349341(n) = { my(f = factor(n)); prod(i=1, #f~, if(2==f[i,1], -1, if(1==f[i,2], -f[i,1], 0))); };

from sympy import prevprime, factorint, prod
def f(p, e):
    return -1 if p == 2 else 0 if e > 1 else -p
def a(n):
    return prod(f(p, e) for p, e in factorint(n).items()) # Sebastian Karlsson, Nov 15 2021

a(1) = 1; a(n) = -Sum_{d|n, d < n} A026741(n/d) * a(d).
a(n) = A349342(n) - A026741(n).
a(2n+1) = A349343(2n+1) for all n >= 1.
Multiplicative with a(2^e) = -1, a(p) = -p and a(p^e) = 0 if e > 1. - Sebastian Karlsson, Nov 15 2021

A349343 Dirichlet inverse of A193356, which is defined as n if n is odd, 0 if n is even.

Original entry on oeis.org

1, 0, -3, 0, -5, 0, -7, 0, 0, 0, -11, 0, -13, 0, 15, 0, -17, 0, -19, 0, 21, 0, -23, 0, 0, 0, 0, 0, -29, 0, -31, 0, 33, 0, 35, 0, -37, 0, 39, 0, -41, 0, -43, 0, 0, 0, -47, 0, 0, 0, 51, 0, -53, 0, 55, 0, 57, 0, -59, 0, -61, 0, 0, 0, 65, 0, -67, 0, 69, 0, -71, 0, -73, 0, 0, 0, 77, 0, -79, 0, 0, 0, -83, 0, 85, 0, 87, 0, -89

Offset: 1

Antti Karttunen, Nov 15 2021

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

Agrees with A349341 on odd numbers.

Cf. A193356, and also A328203, A349134, A349344, A349346, A349353.

a[1]=1;a[n_]:=-DivisorSum[n,If[OddQ[n/#],n/#,0]*a@#&,#Giorgos Kalogeropoulos, Nov 15 2021 *)
f[p_, e_] := If[e == 1, -p, 0]; f[2, e_] := 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2023 *)

A349343(n) = { my(f = factor(n)); prod(i=1, #f~, if((2==f[i,1])||(f[i,2]>1), 0, -f[i,1])); };

a(2n) = 0, a(2n+1) = A349341(2n+1) for all n >= 1.

Multiplicative with a(p^e) = 0 if p=2 or e>1, otherwise a(p) = -p. - (After Sebastian Karlsson's similar formula for A349341).

A349347 Sum of A181988 and its Dirichlet inverse, where A181988(n) = A001511(n)*A003602(n).

Original entry on oeis.org

2, 0, 0, 4, 0, 8, 0, 4, 4, 12, 0, 4, 0, 16, 12, 5, 0, 12, 0, 6, 16, 24, 0, 8, 9, 28, 12, 8, 0, 8, 0, 6, 24, 36, 24, 14, 0, 40, 28, 12, 0, 12, 0, 12, 26, 48, 0, 10, 16, 34, 36, 14, 0, 32, 36, 16, 40, 60, 0, 28, 0, 64, 36, 7, 42, 20, 0, 18, 48, 24, 0, 20, 0, 76, 46, 20, 48, 24, 0, 15, 37, 84, 0, 38, 54, 88, 60, 24, 0, 40

Offset: 1

Antti Karttunen, Nov 15 2021

nonn

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

Cf. A001511, A003602, A181988, A349346.

up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA001511(n) = 1+valuation(n,2);
A003602(n) = (1+(n>>valuation(n,2)))/2;
A181988(n) = (A001511(n)*A003602(n));
v349346 = DirInverseCorrect(vector(up_to,n,A181988(n)));
A349346(n) = v349346[n];
A349347(n) = (A181988(n)+A349346(n));

a(n) = A181988(n) + A349346(n).

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A181988(d) * A349346(n/d).

cp's OEIS Frontend

A349344 Dirichlet inverse of A109168, where A109168(n) = (n+A006519(n))/2, and A006519 is the highest power of 2 dividing n.

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A349341 Dirichlet inverse of A026741, which is defined as n if n is odd, n/2 if n is even.

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A349343 Dirichlet inverse of A193356, which is defined as n if n is odd, 0 if n is even.

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Mathematica

PARI

Formula

A349347 Sum of A181988 and its Dirichlet inverse, where A181988(n) = A001511(n)*A003602(n).

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