A328847 -id:A328847 - 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).

A328848 Number of terms in Zeckendorf expansion needed to write the second Fibonacci based variant of arithmetic derivative of n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 29 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..20000

Cf. A000045, A007895, A324905, A328846, A328847.

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]));
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
A328848(n) = A007895(A328846(n));

a(n) = A007895(A328846(n)).

A374201 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A048679(A328845(i))) = A278222(A048679(A328845(j))), for all i, j >= 1, where A328845 is a Fibonacci-based variant of the arithmetic derivative.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 3, 2, 4, 4, 5, 2, 4, 2, 6, 7, 4, 2, 6, 2, 8, 9, 10, 2, 4, 7, 8, 6, 11, 2, 8, 2, 11, 8, 12, 7, 4, 2, 12, 13, 7, 2, 11, 2, 8, 8, 13, 2, 7, 10, 7, 8, 14, 2, 15, 7, 13, 12, 10, 2, 8, 2, 12, 7, 10, 13, 5, 2, 16, 17, 5, 2, 7, 2, 13, 7, 15, 16, 18, 2, 7, 18, 12, 2, 8, 19, 20, 13, 7, 2, 8, 18, 16, 12, 10, 21, 13, 2, 8, 9, 16

Offset: 0

Antti Karttunen, Jul 02 2024

nonn

Restricted growth sequence transform of A278222(A048679(A328845(n))), or equally, of A304101(A328845(n)).
Related to the Zeckendorf-representation (A014417) of A328845(n).
For all i, j >= 0: a(i) = a(j) => A328847(i) = A328847(j).

Antti Karttunen, Table of n, a(n) for n = 0..75025

Cf. A000045, A014417, A048679, A278222, A304101, A328845, A328847.

Cf. also A304105, A373982.

up_to = 75025;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A106151(n) = { my(s=0, i=0); while(n, if(2!=(n%4), s += (n%2)<>= 1); (s); };
A048679(n) = if(!n,n,A106151(2*A003714(n)));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
A278222(n) = A046523(A005940(1+n));
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]));
v374201 = rgs_transform(vector(1+up_to, n, A278222(A048679(A328845(n-1)))));
A374201(n) = v374201[1+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

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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A328848 Number of terms in Zeckendorf expansion needed to write the second Fibonacci based variant of arithmetic derivative of n.

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A374201 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A048679(A328845(i))) = A278222(A048679(A328845(j))), for all i, j >= 1, where A328845 is a Fibonacci-based 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

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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A328848 Number of terms in Zeckendorf expansion needed to write the second Fibonacci based variant of arithmetic derivative of n.

Views

Author

Keywords

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Crossrefs

Programs

PARI

Formula

A374201 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A048679(A328845(i))) = A278222(A048679(A328845(j))), for all i, j >= 1, where A328845 is a Fibonacci-based 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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Comments

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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.