A347126 -id:A347126 - cp's OEIS frontend

A347129 a(n) = A347130(n) / A003557(n), where A347130 is the Dirichlet convolution of the identity function with the arithmetic derivative of n.

0, 1, 1, 3, 1, 10, 1, 6, 3, 14, 1, 24, 1, 18, 16, 10, 1, 21, 1, 36, 20, 26, 1, 44, 3, 30, 6, 48, 1, 124, 1, 15, 28, 38, 24, 45, 1, 42, 32, 68, 1, 164, 1, 72, 39, 50, 1, 70, 3, 27, 40, 84, 1, 36, 32, 92, 44, 62, 1, 276, 1, 66, 51, 21, 36, 244, 1, 108, 52, 236, 1, 78, 1, 78, 33, 120, 36, 284, 1, 110, 10, 86, 1, 372, 44

Offset: 1

Antti Karttunen, Aug 23 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10080
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003415, A003557, A347126 [= a(A276086(n))], A347130.

Cf. also A342001, A347127, A347128.

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]));
A347130(n) = sumdiv(n,d,d*A003415(n/d));
A347129(n) = (A347130(n) / A003557(n));

a(n) = A347130(n) / A003557(n).

A347396 a(n) = A347395(A276086(n)), where A347395 is Dirichlet convolution of Liouville's lambda with A342001.

Original entry on oeis.org

0, 1, 1, 3, 1, 2, 1, 5, 6, 14, 5, 9, 1, 2, 3, 5, 2, 3, 2, 6, 8, 16, 6, 10, 2, 3, 5, 7, 3, 4, 1, 7, 8, 20, 7, 13, 10, 34, 44, 92, 34, 58, 7, 13, 20, 32, 13, 19, 16, 40, 56, 104, 40, 64, 13, 19, 32, 44, 19, 25, 1, 2, 3, 5, 2, 3, 5, 9, 14, 22, 9, 13, 2, 3, 5, 7, 3, 4, 6, 10, 16, 24, 10, 14, 3, 4, 7, 9, 4, 5, 2, 8, 10, 22

Offset: 0

Antti Karttunen, Sep 02 2021

The scatter plot looks quite peculiar. - Antti Karttunen, Sep 20 2021

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

Cf. A003415, A003557, A008836, A276086, A342001, A347395.

Cf. also A342002, A346471, A347126, A347232, A347962.

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));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A347396(n) = A347395(A276086(n));

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)).

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

A347129 a(n) = A347130(n) / A003557(n), where A347130 is the Dirichlet convolution of the identity function with the arithmetic derivative of n.

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A347396 a(n) = A347395(A276086(n)), where A347395 is Dirichlet convolution of Liouville's lambda with A342001.

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PARI

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

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Programs

PARI

Formula

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

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Programs

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Formula