A328848 -id:A328848 - cp's OEIS frontend

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.

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).

A374209 Number of terms in Zeckendorf representation needed to write A113177(n), where A113177 is fully additive with a(p) = Fibonacci(p).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 02 2024

nonn

Indices for the first occurrences of k=0..6 are: 1, 2, 9, 63, 693, 7623, 105105.

The claim a(n) <= bigomega(n) is true because A007895(n) is the minimum number of Fibonacci numbers which sum to n, regardless of adjacency or duplication. See Apr 17 2015 comments there.

Antti Karttunen, Table of n, a(n) for n = 1..129591
Eric Weisstein's World of Mathematics, Fibonacci Number.
Wikipedia, Zeckendorf's theorem.

Cf. A000045, A001222, A007895, A030426, A113177.

Cf. also A328847, A328848.

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
A113177(n) = if(n<=1, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])));
A374209(n) = if(isprime(n), 1, A007895(A113177(n)));

a(n) = A007895(A113177(n)).
a(p) = 1 for all primes p.
a(n) <= A001222(n), see comments.

cp's OEIS Frontend

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.

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A374209 Number of terms in Zeckendorf representation needed to write A113177(n), where A113177 is fully additive with a(p) = Fibonacci(p).

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cp's OEIS Frontend

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.

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PARI

Formula

A374209 Number of terms in Zeckendorf representation needed to write A113177(n), where A113177 is fully additive with a(p) = Fibonacci(p).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

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.