A349341 -id:A349341 - cp's OEIS frontend

A323910 Dirichlet inverse of the deficiency of n, A033879.

1, -1, -2, 0, -4, 4, -6, 0, -1, 6, -10, 2, -12, 8, 10, 0, -16, 1, -18, 2, 14, 12, -22, 4, -3, 14, -2, 2, -28, -16, -30, 0, 22, 18, 26, 4, -36, 20, 26, 4, -40, -24, -42, 2, 4, 24, -46, 8, -5, -1, 34, 2, -52, 0, 42, 4, 38, 30, -58, 2, -60, 32, 6, 0, 50, -40, -66, 2, 46, -40, -70, 12, -72, 38, 2, 2, 62, -48, -78, 8, -4, 42, -82, -2, 66, 44, 58, 4, -88, 2, 74, 2

Offset: 1

Antti Karttunen, Feb 12 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000
Jon Maiga, Computer-generated formulas for A323910, Sequence Machine.

Cf. A033879, A323911, A323912, A359549 (parity of terms).

Sequences that appear in the convolution formulas: A002033, A008683, A023900, A055615, A046692, A067824, A074206, A174725, A191161, A327960, A328722, A330575, A345182, A349341, A346246, A349387.

b[n_] := 2 n - DivisorSigma[1, n];
a[n_] := a[n] = If[n == 1, 1, -Sum[b[n/d] a[d], {d, Most@ Divisors[n]}]];
Array[a, 100] (* Jean-François Alcover, Feb 17 2020 *)

up_to = 16384;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA033879(n) = (2*n-sigma(n));
v323910 = DirInverse(vector(up_to,n,A033879(n)));
A323910(n) = v323910[n];

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA033879(n/d) * a(d).
From Antti Karttunen, Nov 14 2024: (Start)
Following convolution formulas have been conjectured for this sequence by Sequence Machine, with each one giving the first 10000 terms correctly:
a(n) = Sum_{d|n} A046692(d)*A067824(n/d).
a(n) = Sum_{d|n} A055615(d)*A074206(n/d).
a(n) = Sum_{d|n} A023900(d)*A174725(n/d).
a(n) = Sum_{d|n} A008683(d)*A323912(n/d).
a(n) = Sum_{d|n} A191161(d)*A327960(n/d).
a(n) = Sum_{d|n} A328722(d)*A330575(n/d).
a(n) = Sum_{d|n} A345182(d)*A349341(n/d).
a(n) = Sum_{d|n} A346246(d)*A349387(n/d).
a(n) = Sum_{d|n} A002033(d-1)*A055615(n/d).
(End)

A349353 Dirichlet inverse of A328203.

Original entry on oeis.org

1, -2, -5, 0, -8, 10, -11, 0, 5, 16, -17, 0, -20, 22, 38, 0, -26, -10, -29, 0, 52, 34, -35, 0, 11, 40, 1, 0, -44, -76, -47, 0, 80, 52, 82, 0, -56, 58, 94, 0, -62, -104, -65, 0, -34, 70, -71, 0, 19, -22, 122, 0, -80, -2, 126, 0, 136, 88, -89, 0, -92, 94, -46, 0, 148, -160, -101, 0, 164, -164, -107, 0, -110, 112, -45

Offset: 1

Antti Karttunen, Nov 15 2021

sign

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

Dirichlet convolution of A349134 with A349341, or equally of A349343 with A349344.

Cf. A328203, A349354.

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(dA328203(n) = if(n%2,(1/2)*(sigma(n)+(n*numdiv(n))),2*A328203(n/2));
v349353 = DirInverseCorrect(vector(up_to,n,A328203(n)));
A349353(n) = v349353[n];

a(1) = 1; a(n) = -Sum_{d|n, d < n} A328203(n/d) * a(d).
a(n) = A349354(n) - A328203(n).
a(n) = Sum_{d|n} A349134(d) * A349341(n/d).
a(n) = Sum_{d|n} A349343(d) * A349344(n/d).

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).

A349342 Sum of A026741 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 1, 0, 6, 0, 3, 9, 10, 0, 9, 0, 14, 30, 7, 0, 9, 0, 15, 42, 22, 0, 15, 25, 26, 27, 21, 0, 0, 0, 15, 66, 34, 70, 18, 0, 38, 78, 25, 0, 0, 0, 33, 45, 46, 0, 27, 49, 25, 102, 39, 0, 27, 110, 35, 114, 58, 0, 15, 0, 62, 63, 31, 130, 0, 0, 51, 138, 0, 0, 36, 0, 74, 75, 57, 154, 0, 0, 45, 81, 82, 0, 21, 170, 86

Offset: 1

Antti Karttunen, Nov 15 2021

nonn

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

Cf. A026741, A349341.

A026741(n) = if(n%2,n,n/2);
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))); };
A349342(n) = (A026741(n)+A349341(n));

a(n) = A026741(n) + A349341(n).

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A026741(d) * A349341(n/d).

cp's OEIS Frontend

A323910 Dirichlet inverse of the deficiency of n, A033879.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A349353 Dirichlet inverse of A328203.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A349342 Sum of A026741 and its Dirichlet inverse.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula