A113175 -id:A113175 - cp's OEIS frontend

A324907 a(n) = A007895(A113175(n)).

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

Offset: 1

Antti Karttunen, Apr 14 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..17711

Cf. A113175, A007895, A300837, A324888, A324905.

A113175(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = fibonacci(f[i, 1])); factorback(f); };
A007895(n) = { my(s=0); while(n>0, s++; n -= fibonacci(1+A072649(n))); (s); }
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A324907(n) = A007895(A113175(n));

a(n) = A007895(A113175(n)).

a(2n) = a(n).

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.

Original entry on oeis.org

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).

A113176 Product_{p|n} F(p), where F(p) is the p-th Fibonacci number and where the product is over the distinct prime divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 5, 2, 13, 1, 2, 5, 89, 2, 233, 13, 10, 1, 1597, 2, 4181, 5, 26, 89, 28657, 2, 5, 233, 2, 13, 514229, 10, 1346269, 1, 178, 1597, 65, 2, 24157817, 4181, 466, 5, 165580141, 26, 433494437, 89, 10, 28657, 2971215073, 2, 13, 5, 3194, 233

Offset: 1

Leroy Quet, Oct 16 2005

			63 = 3^2 * 7^1. So a(63) = F(3) * F(7) = 2 * 13 = 26.

Danny Rorabaugh, Table of n, a(n) for n = 1..4000

Cf. A113175, A000045.

b[t_]:=Fibonacci[First[t]] a[n_]:=Apply[Times, Map[b, FactorInteger[n]]] (* Esa Peuha, Oct 26 2005 *)
Table[Times@@(Fibonacci[#]&/@FactorInteger[n][[All,1]]),{n,60}] (* Harvey P. Dale, May 31 2019 *)

[1]+[prod([fibonacci(p) for p in prime_factors(n)]) for n in range(2,53)] # Danny Rorabaugh, Apr 03 2015

Multiplicative with a(p^e) = F(p). - Franklin T. Adams-Watters, Jun 05 2006

More terms from Esa Peuha (esa.peuha(AT)helsinki.fi), Oct 26 2005

cp's OEIS Frontend

A324907 a(n) = A007895(A113175(n)).

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.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A113176 Product_{p|n} F(p), where F(p) is the p-th Fibonacci number and where the product is over the distinct prime divisors of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

Extensions

cp's OEIS Frontend

A324907 a(n) = A007895(A113175(n)).

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(u*v) = a(u)*v + u*a(v), with a(0) = a(1) = 0.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A113176 Product_{p|n} F(p), where F(p) is the p-th Fibonacci number and where the product is over the distinct prime divisors of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

Extensions

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.