A340140 -id:A340140 - cp's OEIS frontend

A340197 a(1) = 1; for n > 1, a(n) = Sum_{d|n, dA318833(n/d)-1) * A340197(d).

1, 0, 0, 2, 0, 7, 0, 6, 6, 13, 0, 13, 0, 19, 22, 18, 0, 19, 0, 23, 32, 31, 0, 53, 20, 37, 24, 33, 0, 21, 0, 54, 52, 49, 58, 110, 0, 55, 62, 95, 0, 29, 0, 53, 52, 67, 0, 185, 42, 53, 82, 63, 0, 139, 94, 137, 92, 85, 0, 321, 0, 91, 74, 162, 112, 45, 0, 83, 112, 45, 0, 403, 0, 109, 82, 93, 136, 53, 0, 331, 114, 121

Offset: 1

Antti Karttunen, Jan 05 2021

nonn

See comments and question in A340140.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A023900, A318833, A340140, A340367.

A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
A318833(n) = (n+A023900(n));
A340197(n) = if(1==n,1,sumdiv(n,d,if(dA318833(n/d)-1)*A340197(d),0)));

a(1) = 1; for n > 1, a(n) = Sum_{d|n, dA318833(n/d)-1) * A340197(d).

A340367 Dirichlet inverse of sequence b(n) = 1-A318833(n).

Original entry on oeis.org

-1, 0, 0, 2, 0, 7, 0, 6, 6, 13, 0, 13, 0, 19, 22, 10, 0, 19, 0, 23, 32, 31, 0, -3, 20, 37, 24, 33, 0, 21, 0, 6, 52, 49, 58, -36, 0, 55, 62, -9, 0, 29, 0, 53, 52, 67, 0, -87, 42, 53, 82, 63, 0, -29, 94, -15, 92, 85, 0, -219, 0, 91, 74, -22, 112, 45, 0, 83, 112, 45, 0, -257, 0, 109, 82, 93, 136, 53, 0, -165, 42, 121

Offset: 1

Antti Karttunen, Jan 05 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A023900, A318833, A340198.

Cf. also A340140, A340197 for similar definitions.

up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
A318833(n) = (n+A023900(n));
v340367 = DirInverseCorrect(vector(up_to, n, 1-A318833(n)));
A340367(n) = v340367[n];

\\ Or as:
A340367(n) = if(1==n, -1, sumdiv(n, d, if(dA318833(n/d))*A340367(d), 0)));

a(1) = -1, for n > 1, a(n) = Sum_{d|n, dA318833(n/d)) * a(d).

A340090 Dirichlet inverse of A219428, n - phi(n) - 1.

Original entry on oeis.org

-1, 0, 0, -1, 0, -3, 0, -3, -2, -5, 0, -7, 0, -7, -6, -8, 0, -11, 0, -11, -8, -11, 0, -21, -4, -13, -8, -15, 0, -21, 0, -21, -12, -17, -10, -36, 0, -19, -14, -33, 0, -29, 0, -23, -20, -23, 0, -63, -6, -29, -18, -27, 0, -47, -14, -45, -20, -29, 0, -85, 0, -31, -26, -55, -16, -45, 0, -35, -24, -45, 0, -123, 0, -37

Offset: 1

Antti Karttunen, Jan 05 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A033987, A051953, A219428, A340094, A340140, A340197, A340198.

up_to = 2^14;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA219428(n) = (n - 1 - eulerphi(n));
v340090 = DirInverseCorrect(vector(up_to, n, A219428(n)));
A340090(n) = v340090[n];
\\ Or as:
A340090(n) = if(1==n, -1, sumdiv(n, d, if(dA219428(n/d)*A340090(d), 0)));

a(1) = -1, for n > 1, a(n) = Sum_{d|n, dA219428(n/d) * a(d).

cp's OEIS Frontend

A340197 a(1) = 1; for n > 1, a(n) = Sum_{d|n, dA318833(n/d)-1) * A340197(d).

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A340367 Dirichlet inverse of sequence b(n) = 1-A318833(n).

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A340090 Dirichlet inverse of A219428, n - phi(n) - 1.

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