A340094 -id:A340094 - cp's OEIS frontend

A340384 Positions of seventeens in A340094, the Dirichlet inverse of f(n) = n - phi(n) + 1.

42, 70, 154, 176, 182, 238, 266, 322, 406, 434, 518, 574, 602, 658, 742, 826, 854, 938, 994, 1022, 1106, 1162, 1246, 1250, 1358, 1414, 1442, 1498, 1526, 1582, 1664, 1778, 1834, 1918, 1946, 2086, 2114, 2198, 2282, 2338, 2422, 2506, 2534, 2674, 2702, 2758, 2786, 2954, 3122, 3178, 3206, 3262, 3346, 3374, 3514, 3598

Offset: 1

Antti Karttunen, Jan 07 2021

nonn

These terms correspond to the horizontal line shown above the X-axis in the logarithmic scatter plots of A340094.

The only non-multiples of 7 among the first 13832 terms are 176, 1250, 1664. Apart from such exceptions (are there more?), all other terms are of the form 2*7*p, with p running through all odd primes in A065091 \ {7}.

Antti Karttunen, Table of n, a(n) for n = 1..13832; all terms < 2^21

Cf. A065091, A340094.

PARI
```
isA340384(n) = (17==A340094(n));
```

{n such that A340094(n) = 17}.

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).

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

A340384 Positions of seventeens in A340094, the Dirichlet inverse of f(n) = n - phi(n) + 1.

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

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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.

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