A340090 -id:A340090 - cp's OEIS frontend

A340094 Dirichlet inverse of A062830, n - phi(n) + 1.

1, -2, -2, 1, -2, 3, -2, -1, 0, 1, -2, -1, -2, -1, 0, 0, -2, -1, -2, 3, -2, -5, -2, 1, -2, -7, -2, 7, -2, 9, -2, -1, -6, -11, -4, 4, -2, -13, -8, -3, -2, 17, -2, 15, 2, -17, -2, 1, -4, -3, -12, 19, -2, 3, -8, -7, -14, -23, -2, -11, -2, -25, 4, -1, -10, 33, -2, 27, -18, 17, -2, -3, -2, -31, -4, 31, -10, 41, -2, 5

Offset: 1

Antti Karttunen, Jan 05 2021

sign

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

Cf. A000010, A051953, A062830, A340090.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1])*sumdiv(n, d, if(dA062830(n) = 1+(n-eulerphi(n));
v340094 = DirInverseCorrect(vector(up_to, n, A062830(n)));
A340094(n) = v340094[n];
\\ Or as:
A340094(n) = if(1==n, 1, -sumdiv(n, d, if(dA062830(n/d)*A340094(d), 0)));

a(1) = 1, for n > 1, a(n) = -Sum_{d|n, dA062830(n/d) * a(d).

A340198 Dirichlet inverse of sequence f(n) = A319340(n)-1 = (A000010(n) + A023900(n) - 1), where A000010 is Euler Totient function phi, and A023900 is its Dirichlet inverse.

Original entry on oeis.org

1, 1, 1, 1, 1, -1, 1, -1, -2, -5, 1, -8, 1, -9, -13, -9, 1, -16, 1, -22, -21, -17, 1, -28, -14, -21, -20, -36, 1, -43, 1, -31, -37, -29, -45, -49, 1, -33, -45, -62, 1, -67, 1, -64, -64, -41, 1, -69, -34, -64, -61, -78, 1, -68, -77, -96, -69, -53, 1, -88, 1, -57, -96, -79, -93, -115, 1, -106, -85, -123, 1, -95, 1, -69

Offset: 1

Antti Karttunen, Jan 05 2021

Conversely, the Dirichlet inverse of this sequence yields a sequence which is one less than A319340, i.e., pointwise sum s(n) = A109606(n) + A023900(n).

a(9796) = 0 is the only zero among the first 2^22 terms.

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

Cf. A000010, A023900, A109606, A319340, A340090, A340094, A340197, A340367.

A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
A319340(n) = (eulerphi(n)+A023900(n));
A340198(n) = if(1==n,1,-sumdiv(n,d,if(dA319340(n/d)-1)*A340198(d),0)));

a(1) = 1, for n > 1, a(n) = -Sum_{d|n, dA319340(n/d)-1) * a(d).

cp's OEIS Frontend

A340094 Dirichlet inverse of A062830, n - phi(n) + 1.

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