A374203 -id:A374203 - cp's OEIS frontend

A328845 The first Fibonacci based variant of arithmetic derivative: a(p) = A000045(p) for prime p, a(uv) = a(u)v + u*a(v), with a(0) = a(1) = 0.

0, 0, 1, 2, 4, 5, 7, 13, 12, 12, 15, 89, 20, 233, 33, 25, 32, 1597, 33, 4181, 40, 53, 189, 28657, 52, 50, 479, 54, 80, 514229, 65, 1346269, 80, 289, 3211, 100, 84, 24157817, 8381, 725, 100, 165580141, 127, 433494437, 400, 105, 57337, 2971215073, 128, 182, 125, 4825, 984, 53316291173, 135, 500, 188, 12581, 1028487, 956722026041, 160

Offset: 0

Antti Karttunen, Oct 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1001

Cf. A000040, A000045, A007895, A030426, A113175, A328847, A374201.
Cf. A374046 (indices of even terms), A374047 (of odd terms), A374122 (of multiples of 3), A374202 (2-adic valuation), A374203 (3-adic valuation), A374205 (5-adic valuation), A374125 [a(n) mod 360].
Cf. A374106 [gcd(a(n), A113177(n))], A374035 [gcd(a(n), A328846(n))], A374116 [gcd(a(n), A328768(n))].
For variants of the same formula, see A003415, A258851, A328768, A328769, A328846, A371192.

A328845[n_] := If[n <= 1, 0, n*Total[MapApply[#2*Fibonacci[#]/# &, FactorInteger[n]]]];
Array[A328845, 100, 0] (* Paolo Xausa, Dec 16 2024 *)

A328845(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])/f[i, 1]));

a(n) = n * Sum e_j * A000045(p_j)/p_j for n = Product p_j^e_j.
a(A000040(n)) = A030426(n).
A007895(a(n)) = A328847(n).

A374202 The 2-adic valuation of A328845(n), where A328845 is a Fibonacci-based variant of the arithmetic derivative.

Original entry on oeis.org

0, 1, 2, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 5, 0, 0, 0, 3, 0, 0, 0, 2, 1, 0, 1, 4, 0, 0, 0, 4, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 7, 1, 0, 0, 3, 0, 0, 2, 2, 0, 0, 0, 5, 0, 0, 0, 6, 1, 0, 0, 3, 0, 0, 0, 2, 0, 0, 3, 5, 1, 0, 0, 4, 3, 0, 0, 3, 1, 0, 0, 2, 0, 0, 3, 5, 0, 0, 3, 4, 0, 0, 0, 2, 0, 0, 0, 2, 1, 0, 0, 2

Offset: 2

Antti Karttunen, Jul 01 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 2..100000

Cf. A007814, A328845, A374045, A374046 (after its 2 initial terms, gives the indices of nonzero terms in this sequence), A374047 (indices of 0's).

Cf. also A374132, A374212, A374203, A374205.

A374202[n_] := IntegerExponent[n*Total[MapApply[#2*Fibonacci[#]/# &, FactorInteger[n]]], 2];
Array[A374202, 100, 2] (* Paolo Xausa, Dec 16 2024 *)

A328845(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])/f[i, 1]));
A374202(n) = valuation(A328845(n), 2);

a(n) = A007814(A328845(n)).

A374207 The 3-adic valuation of A113177(n), where A113177 is fully additive with a(p) = Fibonacci(p).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 2, 1, 1, 0, 0, 0, 0, 0, 0, 2, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 3, 1, 0, 1, 0, 0, 2, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 3, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 2

Antti Karttunen, Jul 01 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 2..100000

Cf. A007949, A113177, A374051, A374052 (indices of nonzero terms).

Cf. also A374133, A374213, A374203, A374206, A374208.

A113177(n) = if(n<=1, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])));
A374207(n) = valuation(A113177(n), 3);

a(n) = A007949(A113177(n)).

cp's OEIS Frontend

A328845 The first Fibonacci based variant of arithmetic derivative: a(p) = A000045(p) for prime p, a(uv) = a(u)v + u*a(v), with a(0) = a(1) = 0.

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A374202 The 2-adic valuation of A328845(n), where A328845 is a Fibonacci-based variant of the arithmetic derivative.

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A374207 The 3-adic valuation of A113177(n), where A113177 is fully additive with a(p) = Fibonacci(p).

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cp's OEIS Frontend

A328845 The first Fibonacci based variant of arithmetic derivative: a(p) = A000045(p) for prime p, a(u*v) = a(u)*v + u*a(v), with a(0) = a(1) = 0.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A374202 The 2-adic valuation of A328845(n), where A328845 is a Fibonacci-based variant of the arithmetic derivative.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A374207 The 3-adic valuation of A113177(n), where A113177 is fully additive with a(p) = Fibonacci(p).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A328845 The first Fibonacci based variant of arithmetic derivative: a(p) = A000045(p) for prime p, a(uv) = a(u)v + u*a(v), with a(0) = a(1) = 0.