A347094 -id:A347094 - cp's OEIS frontend

A347093 Sum of A322577 (convolution of Dedekind psi with Euler phi) and its Dirichlet inverse.

2, 0, 0, 16, 0, 48, 0, 24, 36, 80, 0, 36, 0, 112, 120, 73, 0, 64, 0, 60, 168, 176, 0, 192, 100, 208, 96, 84, 0, 0, 0, 156, 264, 272, 280, 336, 0, 304, 312, 320, 0, 0, 0, 132, 160, 368, 0, 378, 196, 192, 408, 156, 0, 432, 440, 448, 456, 464, 0, 960, 0, 496, 224, 373, 520, 0, 0, 204, 552, 0, 0, 688, 0, 592, 288, 228, 616

Offset: 1

Antti Karttunen, Aug 18 2021

nonn

No negative terms in range 1 .. 2^20.

Apparently, A030059 gives the positions of all zeros.

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

Cf. A000010, A001615, A030059, A030229, A322577, A347092, A347094.

up_to = 16384;
A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A322577(n) = sumdiv(n,d,A001615(n/d)*eulerphi(d));
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA322577(n)));
A347092(n) = v347092[n];
A347093(n) = (A322577(n)+A347092(n));

a(n) = A322577(n) + A347092(n).
For n > 1, a(n) = -Sum_{d|n, 1A322577(d) * A347092(n/d).
For all n >= 1, a(A030059(n)) = 0, a(A030229(n)) = 2*A322577(A030229(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).

A347095 Sum of Pillai's arithmetical function (A018804) and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 9, 0, 30, 0, 21, 25, 54, 0, 35, 0, 78, 90, 49, 0, 51, 0, 63, 130, 126, 0, 95, 81, 150, 85, 91, 0, 0, 0, 113, 210, 198, 234, 172, 0, 222, 250, 171, 0, 0, 0, 147, 153, 270, 0, 235, 169, 147, 330, 175, 0, 231, 378, 247, 370, 342, 0, 405, 0, 366, 221, 257, 450, 0, 0, 231, 450, 0, 0, 424, 0, 438, 245, 259, 546

Offset: 1

Antti Karttunen, Aug 18 2021

nonn

No negative terms in range 1 .. 2^20.

Apparently, A030059 gives the positions of all zeros.

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

Cf. A018804, A030059, A030229, A101035.

Cf. also A347094.

up_to = 16384;
A018804(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ From A018804
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA018804(n)));
A101035(n) = v101035[n];
A347095(n) = (A018804(n)+A101035(n));

a(n) = A018804(n) + A101035(n).
For n > 1, a(n) = -Sum_{d|n, 1A018804(d) * A101035(n/d).
For all n >= 1, a(A030059(n)) = 0, a(A030229(n)) = 2*A018804(A030229(n)).

cp's OEIS Frontend

A347093 Sum of A322577 (convolution of Dedekind psi with Euler phi) and its Dirichlet inverse.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

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).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

A347095 Sum of Pillai's arithmetical function (A018804) and its Dirichlet inverse.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula