A347396 -id:A347396 - cp's OEIS frontend

A347395 Dirichlet convolution of Liouville's lambda (A008836) with A342001, where A342001(n) = A003415(n)/A003557(n).

0, 1, 1, 1, 1, 3, 1, 2, 1, 5, 1, 3, 1, 7, 6, 2, 1, 2, 1, 5, 8, 11, 1, 5, 1, 13, 2, 7, 1, 14, 1, 3, 12, 17, 10, 2, 1, 19, 14, 9, 1, 20, 1, 11, 5, 23, 1, 5, 1, 2, 18, 13, 1, 4, 14, 13, 20, 29, 1, 14, 1, 31, 7, 3, 16, 32, 1, 17, 24, 34, 1, 3, 1, 37, 3, 19, 16, 38, 1, 9, 2, 41, 1, 20, 20, 43, 30, 21, 1, 9, 18, 23, 32

Offset: 1

Antti Karttunen, Sep 02 2021

It seems that all the terms after the initial zero are strictly positive. Checked up to n = 2^24. Compare to A346485.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003415, A003557, A008836, A342001, A347396 [= a(A276086(n))].

Cf. also A346485, A347235.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A342001(n) = (A003415(n) / A003557(n));
A008836(n) = ((-1)^bigomega(n));
A347395(n) = sumdiv(n,d,A008836(n/d)*A342001(d));

a(n) = Sum_{d|n} A008836(n/d) * A342001(d).

Sum_{k=1..n} a(k) ~ c * A065464 * Pi^4 * n^2 / 180, where c = Sum_{j>=2} (1/2 + (-1)^j * (Fibonacci(j) - 1/2))*PrimeZetaP(j) = 0.4526952873143153104685540856936425315834753528741817723313791528384... - Vaclav Kotesovec, Mar 04 2023

A347232 a(n) = A346485(A276086(n)), where A346485 is Möbius transform of A342001.

Original entry on oeis.org

0, 1, 1, 3, 1, 1, 1, 5, 6, 14, 4, 4, 1, 1, 2, 2, 0, 0, 1, 1, 2, 2, 0, 0, 1, 1, 2, 2, 0, 0, 1, 7, 8, 20, 6, 6, 10, 34, 44, 92, 24, 24, 6, 6, 12, 12, 0, 0, 6, 6, 12, 12, 0, 0, 6, 6, 12, 12, 0, 0, 1, 1, 2, 2, 0, 0, 4, 4, 8, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 0, 0, 4, 4, 8, 8

Offset: 0

Antti Karttunen, Aug 26 2021

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A003415, A003557, A276086, A342001, A346485.

Cf. also A342002, A346471, A347126, A347396.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A342001(n) = (A003415(n) / A003557(n));
A346485(n) = sumdiv(n,d,moebius(n/d)*A342001(d));
A347232(n) = A346485(A276086(n));

a(n) = A346485(A276086(n)).

A366803 Dirichlet convolution of Liouville's lambda with A342001 applied to Doudna sequence: a(n) = A347395(A005940(1+n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 2, 1, 5, 6, 3, 1, 2, 2, 2, 1, 7, 8, 5, 10, 14, 5, 5, 1, 2, 3, 2, 2, 4, 2, 3, 1, 11, 12, 7, 14, 20, 7, 9, 16, 34, 44, 14, 7, 9, 10, 5, 1, 2, 3, 2, 5, 5, 2, 3, 2, 6, 8, 4, 2, 3, 3, 3, 1, 13, 14, 11, 16, 32, 11, 13, 18, 54, 68, 20, 11, 13, 14, 9, 22, 76, 92, 34, 124, 92, 34, 22, 11, 13, 20, 9, 16

Offset: 0

Antti Karttunen, Oct 26 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A003415, A005940, A008836, A342001, A347395, A366805 (rgs-transform).

Cf. also A347396, A366795, A366801.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A342001(n) = (A003415(n) / A003557(n));
A008836(n) = ((-1)^bigomega(n));
A347395(n) = sumdiv(n,d,A008836(n/d)*A342001(d));
A366803(n) = A347395(A005940(1+n));

A347962 a(n) = A347961(A276086(n)), where A347961 is Dirichlet convolution of A342001 with itself.

Original entry on oeis.org

0, 0, 0, 2, 1, 14, 0, 2, 2, 40, 20, 210, 1, 18, 20, 234, 120, 1040, 4, 52, 60, 576, 300, 2340, 10, 108, 128, 1090, 575, 4170, 0, 2, 2, 48, 24, 266, 2, 56, 60, 808, 410, 3760, 28, 378, 434, 4312, 2240, 17850, 92, 992, 1176, 10128, 5340, 39120, 208, 1954, 2370, 18592, 9920, 68410, 1, 22, 24, 330, 168, 1572, 28, 418

Offset: 0

Antti Karttunen, Sep 25 2021

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A003415, A003557, A276086, A342001, A347961.

Cf. also the scatter plots of A342002, A347396 and A347126 (the most similar to this one).

PARI
```
A347962(n) = A347961(A276086(n));
```

a(n) = A347961(A276086(n)).

cp's OEIS Frontend

A347395 Dirichlet convolution of Liouville's lambda (A008836) with A342001, where A342001(n) = A003415(n)/A003557(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A347232 a(n) = A346485(A276086(n)), where A346485 is Möbius transform of A342001.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A366803 Dirichlet convolution of Liouville's lambda with A342001 applied to Doudna sequence: a(n) = A347395(A005940(1+n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A347962 a(n) = A347961(A276086(n)), where A347961 is Dirichlet convolution of A342001 with itself.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula