A378224 -id:A378224 - cp's OEIS frontend

A378223 Inverse Möbius transform of A345182.

1, 1, 2, 2, 2, 4, 2, 4, 4, 4, 2, 10, 2, 4, 6, 8, 2, 12, 2, 10, 6, 4, 2, 24, 4, 4, 8, 10, 2, 20, 2, 16, 6, 4, 6, 36, 2, 4, 6, 24, 2, 20, 2, 10, 16, 4, 2, 56, 4, 12, 6, 10, 2, 32, 6, 24, 6, 4, 2, 62, 2, 4, 16, 32, 6, 20, 2, 10, 6, 20, 2, 100, 2, 4, 16, 10, 6, 20, 2, 56, 16, 4, 2, 62, 6, 4, 6, 24, 2, 72, 6, 10, 6, 4, 6

Offset: 1

Antti Karttunen, Nov 25 2024

nonn

Apparently the Dirichlet convolution of A002131 and A323910. - Antti Karttunen, Nov 30 2024

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

Cf. A002131, A323910, A345182, A378224 (Dirichlet inverse).
Cf. also A067824.
Odd bisection is not equal to A278223.

memoA345182 = Map();
A345182(n) = if(n<=2, n%2, my(v); if(mapisdefined(memoA345182,n,&v), v, v = sumdiv(n,d,if(dA345182(d),0)); mapput(memoA345182,n,v); (v)));
A378223(n) = sumdiv(n,d,A345182(d));

up_to = 20000;
A378223list(up_to_n) = { my(v=vector(up_to_n)); v[1] = 1; v[2] = 0; for(n=3,up_to_n,v[n] = 1+sumdiv(n,d,(dA378223list(up_to);
A378223(n) = v378223[n];

a(n) = Sum_{d|n} A345182(d).

For n > 2, a(n) = 2*A345182(n).

A378225 Dirichlet inverse of A067824.

Original entry on oeis.org

1, -2, -2, 0, -2, 2, -2, 0, 0, 2, -2, 0, -2, 2, 2, 0, -2, 0, -2, 0, 2, 2, -2, 0, 0, 2, 0, 0, -2, -2, -2, 0, 2, 2, 2, 0, -2, 2, 2, 0, -2, -2, -2, 0, 0, 2, -2, 0, 0, 0, 2, 0, -2, 0, 2, 0, 2, 2, -2, 0, -2, 2, 0, 0, 2, -2, -2, 0, 2, -2, -2, 0, -2, 2, 0, 0, 2, -2, -2, 0, 0, 2, -2, 0, 2, 2, 2, 0, -2, 0, 2, 0, 2, 2, 2, 0, -2

Offset: 1

Antti Karttunen, Nov 25 2024

sign

Möbius transform of A153881.

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

Cf. A008683, A067824, A153881.

Cf. also A378224.

A074206(n) = if(n>1, sumdiv(n, i, if(iA074206(i))), n); \\ From A074206
A067824(n) = sumdiv(n,d,A074206(d));
memoA378225 = Map();
A378225(n) = if(1==n,1,my(v); if(mapisdefined(memoA378225,n,&v), v, v = -sumdiv(n,d,if(dA067824(n/d)*A378225(d),0)); mapput(memoA378225,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA067824(n/d) * a(d).
a(n) = Sum_{d|n} A008683(n/d)*A153881(d).
Dirichlet g.f.: (2 - zeta(s)) / zeta(s). [See Dec 30 2018 formula in A067824]

A378534 Dirichlet convolution of A033879 and A378525.

Original entry on oeis.org

1, -1, -1, -2, -1, -2, -1, 0, -2, -2, -1, 1, -1, -2, -2, 0, -1, 1, -1, 1, -2, -2, -1, 2, -2, -2, 0, 1, -1, 2, -1, 0, -2, -2, -2, 4, -1, -2, -2, 2, -1, 2, -1, 1, 1, -2, -1, 0, -2, 1, -2, 1, -1, 2, -2, 2, -2, -2, -1, 6, -1, -2, 1, 0, -2, 2, -1, 1, -2, 2, -1, -1, -1, -2, 1, 1, -2, 2, -1, 0, 0, -2, -1, 6, -2, -2, -2, 2, -1, 6

Offset: 1

Antti Karttunen, Dec 01 2024

sign

Möbius transform of A378532.

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

Cf. A008683, A033879, A323910, A378532 (inverse Möbius transform), A378533 (Dirichlet inverse), A378542.

Cf. also A378224.

A033879(n) = (n+n-sigma(n));
A378542(n) = sumdiv(n,d,d*!(bigomega(n/d)%2));
memoA378525 = Map();
A378525(n) = if(1==n,1,my(v); if(mapisdefined(memoA378525,n,&v), v, v = -sumdiv(n,d,if(dA378542(n/d)*A378525(d),0)); mapput(memoA378525,n,v); (v)));
A378534(n) = sumdiv(n,d,A033879(d)*A378525(n/d));

a(n) = Sum_{d|n} A033879(d)*A378525(n/d).

a(n) = Sum_{d|n} A008683(d)*A378532(n/d).

cp's OEIS Frontend

A378223 Inverse Möbius transform of A345182.

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A378225 Dirichlet inverse of A067824.

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A378534 Dirichlet convolution of A033879 and A378525.

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