A347128 -id:A347128 - 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).

A348494 a(n) = A348492(n) / A003557(n), where A348492 is the GCD of the arithmetic derivative (A003415) and Pillai's arithmetical function (A018804).

Original entry on oeis.org

1, 1, 1, 2, 1, 5, 1, 1, 1, 1, 1, 4, 1, 3, 1, 2, 1, 7, 1, 12, 5, 1, 1, 1, 1, 15, 3, 4, 1, 1, 1, 1, 7, 1, 3, 2, 1, 3, 1, 1, 1, 1, 1, 12, 1, 5, 1, 2, 1, 3, 5, 4, 1, 9, 1, 1, 1, 1, 1, 2, 1, 3, 1, 2, 9, 1, 1, 12, 1, 1, 1, 1, 1, 3, 1, 4, 3, 1, 1, 2, 1, 1, 1, 2, 11, 15, 1, 35, 1, 1, 5, 12, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Oct 21 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003415, A003557, A018804, A342001, A347128, A348492, A348500 [= a(A276086(n))].

Cf. also A348496.

Array[GCD[Total@ GCD[#1, Range[#1]], #1 Total[#2/#1 & @@@ #2]]/Apply[Times, Map[#1^(#2 - 1) & @@ # &, #2]] & @@ {#, FactorInteger[#]} &, 105] (* Michael De Vlieger, Oct 21 2021 *)

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]));
A018804(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ From A018804
A348492(n) = gcd(A003415(n), A018804(n));
A348494(n) = (A348492(n)/A003557(n));

a(n) = gcd(A342001(n), A347128(n)).

a(n) = A348492(n) / A003557(n), where A348492(n) = gcd(A003415(n), A018804(n)).

A348496 a(n) = gcd(A018804(n), A347130(n)) / A003557(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 4, 1, 3, 1, 2, 1, 21, 1, 36, 5, 1, 1, 1, 1, 15, 3, 4, 1, 1, 1, 1, 7, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 12, 3, 5, 1, 10, 1, 3, 5, 4, 1, 9, 1, 1, 1, 1, 1, 12, 1, 3, 1, 1, 9, 1, 1, 12, 1, 1, 1, 1, 1, 3, 1, 4, 3, 1, 1, 2, 1, 1, 1, 4, 11, 15, 1, 35, 1, 3, 5, 36, 1, 1, 3, 1, 1, 3, 3, 1, 1

Offset: 1

Antti Karttunen, Oct 21 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A003557, A018804, A347128, A347129, A347130, A348495, A348501 [= a(A276086(n))].

Cf. also A348494, A348498.

Table[GCD[Total@ GCD[n, Range[n]], DivisorSum[n, #*(If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &[n/#]) &]]/Apply[Times, Map[#1^(#2 - 1) & @@ # &, FactorInteger[n]]], {n, 101}] (* Michael De Vlieger, Oct 21 2021 *)

\\ Needs also code from A348495:
A003557(n) = (n/factorback(factorint(n)[, 1]));
A348496(n) = (A348495(n)/A003557(n));

a(n) = A348495(n) / A003557(n).

a(n) = gcd(A347128(n), A347129(n)).

A347127 a(n) = A327251(n) / A003557(n).

Original entry on oeis.org

1, 5, 7, 8, 11, 35, 15, 11, 11, 55, 23, 56, 27, 75, 77, 14, 35, 55, 39, 88, 105, 115, 47, 77, 17, 135, 15, 120, 59, 385, 63, 17, 161, 175, 165, 88, 75, 195, 189, 121, 83, 525, 87, 184, 121, 235, 95, 98, 23, 85, 245, 216, 107, 75, 253, 165, 273, 295, 119, 616, 123, 315, 165, 20, 297, 805, 135, 280, 329, 825, 143, 121

Offset: 1

Antti Karttunen, Aug 23 2021

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A001615, A003557, A327251.

Cf. also A048250, A347128, A347129.

f[p_, e_] := (p + 1)*e + p; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 24 2021 *)

A347127(n) = { my(f=factor(n)); prod(i=1, #f~, ((f[i, 1]+1)*f[i, 2] + f[i, 1])); };

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
A003557(n) = (n/factorback(factorint(n)[, 1]));
A327251(n) = sumdiv(n, d, A001615(n/d)*d);
A347127(n) = (A327251(n) / A003557(n));

Multiplicative with a(p^e) = ((p+1)*e + p) for prime p.

a(n) = A327251(n) / A003557(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.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A348494 a(n) = A348492(n) / A003557(n), where A348492 is the GCD of the arithmetic derivative (A003415) and Pillai's arithmetical function (A018804).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A348496 a(n) = gcd(A018804(n), A347130(n)) / A003557(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A347127 a(n) = A327251(n) / A003557(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula