A371192 -id:A371192 - 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).

A328768 The first primorial based variant of arithmetic derivative: a(prime(i)) = A002110(i-1), where prime(i) = A000040(i), a(uv) = a(u)v + u*a(v), with a(0) = a(1) = 0.

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 7, 30, 12, 12, 17, 210, 20, 2310, 67, 28, 32, 30030, 33, 510510, 44, 104, 431, 9699690, 52, 60, 4633, 54, 148, 223092870, 71, 6469693230, 80, 652, 60077, 192, 84, 200560490130, 1021039, 6956, 108, 7420738134810, 229, 304250263527210, 884, 114, 19399403, 13082761331670030, 128, 420, 145, 90124, 9292, 614889782588491410, 135, 1116, 324

Offset: 0

Antti Karttunen, Oct 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1001
Index entries for sequences related to primorial numbers

Cf. A002110, A108951, A276085, A276150, A328771, A373982, A374211.
Cf. A042965 (indices of even terms), A016825 (of odd terms), A152822 (antiparity of terms), A373992 (indices of multiples of 3), A374212 (2-adic valuation), A374213 (3-adic valuation), A374123 [a(n) mod 360].
Cf. A374031 [gcd(a(n), A276085(n))], A374116 [gcd(a(n), A328845(n))].
For variants of the same formula, see A003415, A258851, A328769, A328845, A328846, A371192.

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]));

A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i, 1]))/(f[i, 1]^2)));

a(n) = n * Sum e_j * A276085(p_j)/p_j for n = Product p_j^e_j, where for primes p, A276085(p) = A002110(A000720(p)-1).
a(n) = n * Sum e_j * (p_j)#/(p_j^2) for n = Product p_j^e_j with (p_j)# = A034386(p_j).
For all n >= 0, A276150(a(n)) = A328771(n).

A328846 The second Fibonacci based variant of arithmetic derivative: a(p) = A000045(2+A000720(p)) for prime p, a(uv) = a(u)v + u*a(v), with a(0) = a(1) = 0. Also called PrimePi-Fibonacci variant of the arithmetic derivative.

Original entry on oeis.org

0, 0, 2, 3, 8, 5, 12, 8, 24, 18, 20, 13, 36, 21, 30, 30, 64, 34, 54, 55, 60, 45, 48, 89, 96, 50, 68, 81, 88, 144, 90, 233, 160, 72, 102, 75, 144, 377, 148, 102, 160, 610, 132, 987, 140, 135, 224, 1597, 240, 112, 150, 153, 188, 2584, 216, 120, 232, 222, 346, 4181, 240, 6765, 528, 198, 384, 170, 210, 10946, 272, 336, 220, 17711, 360

Offset: 0

Antti Karttunen, Oct 28 2019

nonn

Cf. A000040, A000045, A000720, A003965, A007895, A328848.
Cf. also A003415, A258851, A328768, A328769, A328845 for other arithmetic derivatives, and also A371192 for another PrimePi-Fibonacci variant.
Cf. A374035 [= gcd(a(n), A328845(n))], A374048 (antiparity of this sequence), A374049 (indices of even terms), A374050 (of odd terms).

A328846(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*fibonacci(2+primepi(f[i,1]))/f[i, 1]));

a(n) = n * Sum e_j * A000045(2+A000720(p_j))/p_j for n = Product p_j^e_j.
a(A000040(n)) = A000045(2+n).
A007895(a(n)) = A328848(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.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A328768 The first primorial based variant of arithmetic derivative: a(prime(i)) = A002110(i-1), where prime(i) = A000040(i), a(uv) = a(u)v + u*a(v), with a(0) = a(1) = 0.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A328846 The second Fibonacci based variant of arithmetic derivative: a(p) = A000045(2+A000720(p)) for prime p, a(uv) = a(u)v + u*a(v), with a(0) = a(1) = 0. Also called PrimePi-Fibonacci variant of the arithmetic derivative.

Views

Author

Keywords

Links

Crossrefs

Programs

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.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A328768 The first primorial based variant of arithmetic derivative: a(prime(i)) = A002110(i-1), where prime(i) = A000040(i), a(u*v) = a(u)*v + u*a(v), with a(0) = a(1) = 0.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A328846 The second Fibonacci based variant of arithmetic derivative: a(p) = A000045(2+A000720(p)) for prime p, a(u*v) = a(u)*v + u*a(v), with a(0) = a(1) = 0. Also called PrimePi-Fibonacci variant of the arithmetic derivative.

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.

A328768 The first primorial based variant of arithmetic derivative: a(prime(i)) = A002110(i-1), where prime(i) = A000040(i), a(uv) = a(u)v + u*a(v), with a(0) = a(1) = 0.

A328846 The second Fibonacci based variant of arithmetic derivative: a(p) = A000045(2+A000720(p)) for prime p, a(uv) = a(u)v + u*a(v), with a(0) = a(1) = 0. Also called PrimePi-Fibonacci variant of the arithmetic derivative.