A319340 -id:A319340 - cp's OEIS frontend

A340188 Sum of A063994 and its Dirichlet inverse, where A063994(x) = Product_{primes p dividing x} gcd(p-1, x-1).

2, 0, 0, 1, 0, 4, 0, 1, 4, 8, 0, 0, 0, 12, 16, 1, 0, -4, 0, -2, 24, 20, 0, 1, 16, 24, 0, -4, 0, -28, 0, 1, 40, 32, 48, 5, 0, 36, 48, 1, 0, -48, 0, -8, -16, 44, 0, 1, 36, -32, 64, -10, 0, 8, 80, 5, 72, 56, 0, 24, 0, 60, -32, 1, 96, -88, 0, -14, 88, -116, 0, 0, 0, 72, -48, -16, 120, -108, 0, 1, 4, 80, 0, 48, 128, 84, 112

Offset: 1

Antti Karttunen, Dec 31 2020

sign

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A063994, A318828, A340187, A340189.

Cf. also A319340, A340191.

up_to = 65537;
A063994(n) = { my(f=factor(n)); prod(i=1, #f~, gcd(f[i, 1]-1, n-1)); };
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA063994(n)));
A340187(n) = v340187[n];
A340188(n) = (A063994(n)+A340187(n));

a(n) = A063994(n) + A340187(n).

a(n) = A340189(n) - A318828(n).

A354105 Sum of A354102 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 1, 0, 8, 0, 3, 16, 4, 0, 12, 0, 24, 16, 7, 0, 24, 0, 6, 96, 32, 0, 20, 4, 12, 96, 36, 0, 0, 0, 15, 128, 20, 48, 44, 0, 56, 48, 10, 0, 0, 0, 48, 48, 72, 0, 36, 144, 8, 80, 18, 0, 104, 64, 60, 224, 36, 0, 8, 0, 80, 288, 31, 24, 0, 0, 30, 288, 0, 0, 84, 0, 44, 32, 84, 384, 0, 0, 18, 496, 60, 0, 48, 40, 104, 144

Offset: 1

Antti Karttunen, May 18 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A267099, A319340, A354102, A354104.

PARI
```
A354105(n) = (A354102(n)+A354104(n));
```

a(n) = A354102(n) + A354104(n).
a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A354102(d) * A354104(n/d).
a(n) = A319340(A267099(n)).

A346488 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), for all i, j >= 1, where f(n) = 0 if mu(n) = -1, and f(n) = n for all other numbers (with mu = Möbius mu, A008683).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 2, 2, 21, 22, 23, 24, 25, 2, 26, 27, 28, 2, 2, 2, 29, 30, 31, 2, 32, 33, 34, 35, 36, 2, 37, 38, 39, 40, 41, 2, 42, 2, 43, 44, 45, 46, 2, 2, 47, 48, 2, 2, 49, 2, 50, 51, 52, 53, 2, 2, 54, 55, 56, 2, 57, 58, 59, 60, 61, 2, 62, 63, 64, 65, 66, 67, 68, 2, 69, 70, 71

Offset: 1

Antti Karttunen, Aug 20 2021

nonn

Restricted growth sequence transform of the sequence f(n) = 0 if mu(n) = -1, and f(n) = n for mu(n) >= 0.
For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j) => A305980(i) = A305980(j),
  a(i) = a(j) => b(i) = b(j), where b is the pointwise sum of any two multiplicative sequences c and d that are Dirichlet inverses of each other. For example, b can be a sequence like A319340, A323885, or A347094.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A008683, A070549, A030059 (positions of 2's).

Cf. also A305800, A305980, A319340, A323885, A347094.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
Aux346488(n) = if(moebius(n)<0,0,n);
v346488 = rgs_transform(vector(up_to, n, Aux346488(n)));
A346488(n) = v346488[n];

A070549(n) = sum(k=1,n,(-1==moebius(k)));
A346488(n) = if(1==n,1,if(-1==moebius(n),2,1+n-A070549(n)));

a(1) = 1, and for n > 1, if A008683(n) = -1, a(n) = 2, otherwise a(n) = 1 + n - A070549(n).

cp's OEIS Frontend

A340188 Sum of A063994 and its Dirichlet inverse, where A063994(x) = Product_{primes p dividing x} gcd(p-1, x-1).

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A354105 Sum of A354102 and its Dirichlet inverse.

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A346488 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), for all i, j >= 1, where f(n) = 0 if mu(n) = -1, and f(n) = n for all other numbers (with mu = Möbius mu, A008683).

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