A342413 -id:A342413 - cp's OEIS frontend

A342414 a(n) = A003415(n) / gcd(phi(n),A003415(n)), where A003415(n) is the arithmetic derivative of n, and phi is Euler totient function.

0, 1, 1, 2, 1, 5, 1, 3, 1, 7, 1, 4, 1, 3, 1, 4, 1, 7, 1, 3, 5, 13, 1, 11, 1, 5, 3, 8, 1, 31, 1, 5, 7, 19, 1, 5, 1, 7, 2, 17, 1, 41, 1, 12, 13, 25, 1, 7, 1, 9, 5, 7, 1, 9, 2, 23, 11, 31, 1, 23, 1, 11, 17, 6, 3, 61, 1, 9, 13, 59, 1, 13, 1, 13, 11, 20, 3, 71, 1, 11, 2, 43, 1, 31, 11, 15, 4, 7, 1, 41, 5, 24, 17, 49, 1, 17

Offset: 1

Antti Karttunen, Mar 11 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12500
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000010, A003415, A342001, A342008 (positions of ones), A342413, A342415, A342416.

Array[#1/GCD[#1, #2] & @@ {If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &@ Abs[#], EulerPhi[#]} &, 96] (* Michael De Vlieger, Mar 11 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342414(n) = { my(u=A003415(n)); (u/gcd(eulerphi(n),u)); };

a(n) = A003415(n) / A342413(n) = A003415(n) / gcd(A000010(n),A003415(n)).

a(n) = A342001(n) / A342416(n).

A342458 a(n) = gcd(A001615(n), A003415(n)), where A001615 is Dedekind psi, and A003415 is the arithmetic derivative of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 12, 6, 1, 1, 8, 1, 3, 8, 8, 1, 3, 1, 12, 2, 1, 1, 4, 10, 3, 9, 16, 1, 1, 1, 16, 2, 1, 12, 12, 1, 3, 8, 4, 1, 1, 1, 24, 3, 1, 1, 16, 14, 45, 4, 28, 1, 27, 8, 4, 2, 1, 1, 4, 1, 3, 3, 96, 6, 1, 1, 36, 2, 1, 1, 12, 1, 3, 5, 40, 6, 1, 1, 16, 108, 1, 1, 4, 2, 3, 8, 4, 1, 3, 4, 48, 2, 1, 24, 16, 1, 7, 3, 20

Offset: 1

Antti Karttunen, Mar 28 2021

nonn

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

Cf. A001615, A003415, A003557, A342459, A342919.
Cf. A301939 (gives the positions at which a(n) = A001615(n) = A003415(n)).
Cf. also A175732, A342413, A342915.

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
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342458(n) = gcd(A001615(n), A003415(n));

a(n) = gcd(A001615(n), A003415(n)).
a(n) = A003557(n) * A342459(n).
a(n) = A003415(n) / A342919(n).

A342416 a(n) = gcd(A173557(n), A342001(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 3, 8, 1, 1, 1, 1, 4, 2, 1, 1, 1, 2, 3, 1, 2, 1, 1, 1, 1, 2, 1, 12, 2, 1, 3, 8, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 4, 4, 1, 1, 8, 1, 2, 1, 1, 2, 1, 3, 1, 1, 6, 1, 1, 4, 2, 1, 1, 1, 1, 3, 1, 2, 6, 1, 1, 2, 2, 1, 1, 2, 2, 3, 8, 5, 1, 1, 4, 2, 2, 1, 24, 1, 1, 1, 5, 2, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Mar 11 2021

nonn

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

Cf. A000010, A003415, A003557, A173557, A342001, A342413.

Array[GCD[#2, #1/#3] & @@ {If[#1 < 2, 0, #1 Total[#2/#1 & @@@ #2]], If[#1 == 1, 1, Times @@ Map[# - 1 &, #2[[All, 1]] ]], #1/Times @@ #2[[All, 1]]} & @@ {Abs[#], FactorInteger[#]} &, 91] (* Michael De Vlieger, Mar 11 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]));
A342413(n) = gcd(eulerphi(n), A003415(n));
A342416(n) = (A342413(n)/A003557(n));

a(n) = A342413(n) / A003557(n) = gcd(A173557(n), A342001(n)).

A348492 Greatest common divisor of the arithmetic derivative (A003415) and Pillai's arithmetical function (A018804).

Original entry on oeis.org

1, 1, 1, 4, 1, 5, 1, 4, 3, 1, 1, 8, 1, 3, 1, 16, 1, 21, 1, 24, 5, 1, 1, 4, 5, 15, 27, 8, 1, 1, 1, 16, 7, 1, 3, 12, 1, 3, 1, 4, 1, 1, 1, 24, 3, 5, 1, 16, 7, 15, 5, 8, 1, 81, 1, 4, 1, 1, 1, 4, 1, 3, 3, 64, 9, 1, 1, 24, 1, 1, 1, 12, 1, 3, 5, 8, 3, 1, 1, 16, 27, 1, 1, 4, 11, 15, 1, 140, 1, 3, 5, 24, 1, 1, 3, 16, 1, 7

Offset: 1

Antti Karttunen, Oct 21 2021

nonn

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

Cf. A003415, A003557, A018804, A348493, A348494.

Cf. also A342413, A342458, A348495.

Array[GCD[Total@ GCD[#, Range[#]], # Total[#2/#1 & @@@ FactorInteger[#]]] &, 98] (* 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]));
A018804(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ From A018804
A348492(n) = gcd(A003415(n), A018804(n));

a(n) = gcd(A003415(n), A018804(n)).
For n > 1, a(n) = A003415(n) / A348493(n).
a(n) = A003557(n) * A348494(n).

A342415 a(n) = phi(n) / gcd(phi(n),A003415(n)), where A003415(n) is the arithmetic derivative of n, and phi is Euler totient function.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 6, 1, 1, 4, 10, 1, 12, 2, 1, 1, 16, 2, 18, 1, 6, 10, 22, 2, 2, 4, 2, 3, 28, 8, 30, 1, 10, 16, 2, 1, 36, 6, 3, 4, 40, 12, 42, 5, 8, 22, 46, 1, 3, 4, 8, 3, 52, 2, 5, 6, 18, 28, 58, 4, 60, 10, 12, 1, 8, 20, 66, 4, 22, 24, 70, 2, 72, 12, 8, 9, 10, 24, 78, 2, 1, 40, 82, 6, 32, 14, 7, 2, 88, 8, 18, 11, 30

Offset: 1

Antti Karttunen, Mar 11 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12500

Cf. A000010, A003415, A173557, A342009 (positions of ones), A342413, A342414, A342416.

Array[#2/GCD[#1, #2] & @@ {If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &@ Abs[#], EulerPhi[#]} &, 93] (* Michael De Vlieger, Mar 11 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342415(n) = { my(u=eulerphi(n)); (u/gcd(u,A003415(n))); };

a(n) = A000010(n) / A342413(n) = A000010(n) / gcd(A000010(n),A003415(n)).

a(n) = A173557(n) / A342416(n).

A348028 Greatest common divisor of A003415 (arithmetic derivative) and sigma, the sum of divisors function.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 3, 8, 1, 1, 3, 1, 6, 2, 1, 1, 4, 1, 3, 1, 8, 1, 1, 1, 1, 2, 1, 12, 1, 1, 3, 8, 2, 1, 1, 1, 12, 39, 1, 1, 4, 1, 3, 4, 14, 1, 3, 8, 4, 2, 1, 1, 4, 1, 3, 1, 1, 6, 1, 1, 18, 2, 1, 1, 39, 1, 3, 1, 20, 6, 1, 1, 2, 1, 1, 1, 4, 2, 3, 8, 20, 1, 3, 4, 24, 2, 1, 24, 4, 1, 1, 3, 7, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Sep 25 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A003415.

Cf. also A342413, A342458, A343226.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A348028(n) = gcd(sigma(n), A003415(n));

a(n) = gcd(A000203(n), A003415(n)).

cp's OEIS Frontend

A342414 a(n) = A003415(n) / gcd(phi(n),A003415(n)), where A003415(n) is the arithmetic derivative of n, and phi is Euler totient function.

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A342458 a(n) = gcd(A001615(n), A003415(n)), where A001615 is Dedekind psi, and A003415 is the arithmetic derivative of n.

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A342416 a(n) = gcd(A173557(n), A342001(n)).

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A348492 Greatest common divisor of the arithmetic derivative (A003415) and Pillai's arithmetical function (A018804).

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A342415 a(n) = phi(n) / gcd(phi(n),A003415(n)), where A003415(n) is the arithmetic derivative of n, and phi is Euler totient function.

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A348028 Greatest common divisor of A003415 (arithmetic derivative) and sigma, the sum of divisors function.

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Author

Keywords

Links

Crossrefs

Programs

PARI

Formula