A378433 -id:A378433 - cp's OEIS frontend

A378434 Arithmetic mean between the Dirichlet inverses of {sum of unitary divisors} and {sum of squarefree divisors}.

1, -3, -4, 5, -6, 12, -8, -9, 9, 18, -12, -20, -14, 24, 24, 16, -18, -27, -20, -30, 32, 36, -24, 36, 20, 42, -24, -40, -30, -72, -32, -30, 48, 54, 48, 48, -38, 60, 56, 54, -42, -96, -44, -60, -54, 72, -48, -64, 35, -60, 72, -70, -54, 72, 72, 72, 80, 90, -60, 120, -62, 96, -72, 56, 84, -144, -68, -90, 96, -144, -72, -90

Offset: 1

Antti Karttunen, Nov 26 2024

sign

Arithmetic mean between A158523 and A178450.

Apparently differs from A378433 at positions given by A048111: 16, 32, 36, 48, 64, 72, 80, 81, 96, ...

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

Cf. A034448, A048111, A048250, A158523, A178450, A325973, A378433, A378435 (Dirichlet inverse).

A158523(n) = { my(f = factor(n)); prod(i = 1, #f~, (-1)^f[i, 2]*(f[i, 1]+1)*f[i, 1]^(f[i, 2]-1)); }; \\ From A158523
A178450(n) = { my(f=factor(n)); prod(i=1, #f~, if(!(f[i,2]%2), 2*(f[i, 1]^(f[i, 2]/2)), -(1+f[i,1])*(f[i, 1]^((f[i, 2]-1)/2)))); };
A378434(n) = ((A158523(n)+A178450(n))/2);

a(n) = (1/2) * (A158523(n)+A178450(n)).

A378435 Dirichlet inverse of the arithmetic mean between the Dirichlet inverses of {sum of unitary divisors} and {sum of squarefree divisors}.

Original entry on oeis.org

1, 3, 4, 4, 6, 12, 8, 6, 7, 18, 12, 16, 14, 24, 24, 9, 18, 21, 20, 24, 32, 36, 24, 24, 16, 42, 16, 32, 30, 72, 32, 15, 48, 54, 48, 25, 38, 60, 56, 36, 42, 96, 44, 48, 42, 72, 48, 36, 29, 48, 72, 56, 54, 48, 72, 48, 80, 90, 60, 96, 62, 96, 56, 24, 84, 144, 68, 72, 96, 144, 72, 33, 74, 114, 64, 80, 96, 168, 80, 54, 34

Offset: 1

Antti Karttunen, Nov 26 2024

sign

The first negative term is a(2592) = -48.

Apparently differs from A325973 at positions given by A048111: 16, 32, 36, 48, 64, 72, 80, 81, 96, ...

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

Dirichlet inverse of A378434.

Cf. A034448, A048111, A048250, A158523, A178450, A325973, A378433.

A158523(n) = { my(f = factor(n)); prod(i = 1, #f~, (-1)^f[i, 2]*(f[i, 1]+1)*f[i, 1]^(f[i, 2]-1)); }; \\ From A158523
A178450(n) = { my(f=factor(n)); prod(i=1, #f~, if(!(f[i,2]%2), 2*(f[i, 1]^(f[i, 2]/2)), -(1+f[i,1])*(f[i, 1]^((f[i, 2]-1)/2)))); };
A378434(n) = ((A158523(n)+A178450(n))/2);
memoA378435 = Map();
A378435(n) = if(1==n,1,my(v); if(mapisdefined(memoA378435,n,&v), v, v = -sumdiv(n,d,if(dA378434(n/d)*A378435(d),0)); mapput(memoA378435,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA378434(n/d) * a(d).

cp's OEIS Frontend

A378434 Arithmetic mean between the Dirichlet inverses of {sum of unitary divisors} and {sum of squarefree divisors}.

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