A374125 -id:A374125 - 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).

A374124 a(n) = A113177(n) mod 360, where A113177 is fully additive with a(p) = Fibonacci(p).

Original entry on oeis.org

0, 1, 2, 2, 5, 3, 13, 3, 4, 6, 89, 4, 233, 14, 7, 4, 157, 5, 221, 7, 15, 90, 217, 5, 10, 234, 6, 15, 149, 8, 229, 5, 91, 158, 18, 6, 17, 222, 235, 8, 301, 16, 77, 91, 9, 218, 73, 6, 26, 11, 159, 235, 293, 7, 94, 16, 223, 150, 161, 9, 161, 230, 17, 6, 238, 92, 293, 159, 219, 19, 289, 7, 73, 18, 12, 223, 102, 236, 301

Offset: 1

Antti Karttunen, Jun 30 2024

nonn

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

Cf. A000045, A113177.
Cf. A373585 (antiparity of terms), A373586 (indices of even terms), A373587 (of odd terms), A374052 (of multiples of 3).
Cf. also A372576, A374123, A374125.

A113177(n) = if(n<=1, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])));
A374124(n) = (A113177(n)%360);

A374123 a(n) = A328768(n) mod 360, where A328768 is the first primorial based variant of the arithmetic derivative.

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 7, 30, 12, 12, 17, 210, 20, 150, 67, 28, 32, 150, 33, 30, 44, 104, 71, 210, 52, 60, 313, 54, 148, 150, 71, 30, 80, 292, 317, 192, 84, 210, 79, 116, 108, 210, 229, 330, 164, 114, 83, 150, 128, 60, 145, 124, 292, 210, 135, 36, 324, 128, 329, 330, 172, 30, 91, 354, 192, 108, 257, 30, 308, 316, 59, 210

Offset: 0

Antti Karttunen, Jun 30 2024

nonn

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

Cf. A002110, A328768.
Cf. A042965 (indices of even terms), A016825 (of odd terms), A152822 (antiparity of terms), A373992 (indices of multiples of 3).
Cf. also A372576, A374124, A374125.

A002110(n) = prod(i=1,n,prime(i));
A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i,1])-1)/f[i, 1]));
A374123(n) = (A328768(n)%360);

A379115 a(n) = A328845(n) mod 5, where A328845 is the first Fibonacci based variant of arithmetic derivative.

Original entry on oeis.org

0, 0, 1, 2, 4, 0, 2, 3, 2, 2, 0, 4, 0, 3, 3, 0, 2, 2, 3, 1, 0, 3, 4, 2, 2, 0, 4, 4, 0, 4, 0, 4, 0, 4, 1, 0, 4, 2, 1, 0, 0, 1, 2, 2, 0, 0, 2, 3, 3, 2, 0, 0, 4, 3, 0, 0, 3, 1, 2, 1, 0, 1, 4, 1, 2, 0, 1, 3, 1, 2, 0, 4, 4, 3, 1, 0, 0, 1, 4, 1, 0, 1, 3, 2, 1, 0, 2, 0, 4, 4, 0, 0, 0, 4, 3, 0, 4, 2, 3, 3, 0, 1, 1, 2, 0, 0

Offset: 0

Antti Karttunen, Dec 15 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 0..81839

Cf. A010874, A328845, A374125, A374205, A379116 (positions of 0's), A379117 (their characteristic function).

A379115[n_] := If[n <= 1, 0, Mod[n*Total[MapApply[#2*Fibonacci[#]/# &, FactorInteger[n]]], 5]];
Array[A379115, 100, 0] (* Paolo Xausa, Dec 16 2024 *)

A379115(n) = if(n<=1, 0, my(f=factor(n)); (n*sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])/f[i, 1]))%5);

a(n) = A010874(A328845(n)) = A010874(A374125(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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A374124 a(n) = A113177(n) mod 360, where A113177 is fully additive with a(p) = Fibonacci(p).

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PARI

A374123 a(n) = A328768(n) mod 360, where A328768 is the first primorial based variant of the arithmetic derivative.

Views

Author

Keywords

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Programs

PARI

A379115 a(n) = A328845(n) mod 5, where A328845 is the first Fibonacci based variant of arithmetic derivative.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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.

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Author

Keywords

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Crossrefs

Programs

Mathematica

PARI

Formula

A374124 a(n) = A113177(n) mod 360, where A113177 is fully additive with a(p) = Fibonacci(p).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A374123 a(n) = A328768(n) mod 360, where A328768 is the first primorial based variant of the arithmetic derivative.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A379115 a(n) = A328845(n) mod 5, where A328845 is the first Fibonacci based variant of arithmetic derivative.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

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.