A375430 -id:A375430 - cp's OEIS frontend

A375431 The indices of the terms of A375430 in the Fibonacci sequence.

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

Offset: 1

Amiram Eldar, Aug 15 2024

Since 1 appears twice in the Fibonacci sequence (1 = Fibonacci(1) = Fibonacci(2)), its index here is chosen to be 2.

			For n = 8 = 2^3, the dual Zeckendorf representation of 3 is 11, i.e., 3 = Fibonacci(2) + Fibonacci(3). Therefore 8 = 2^(Fibonacci(2) + Fibonacci(3)) = 2^Fibonacci(2) * 2^Fibonacci(3), and a(8) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A051903, A072649, A375429, A375430.

A072649[n_] := Module[{k = 0}, While[Fibonacci[k] <= n, k++]; k-2]; a[n_] := A072649[1 + Max[FactorInteger[n][[;;, 2]]]]; a[1] = 0; Array[a, 100]

A072649(n) = {my(k = 0); while(fibonacci(k) <= n, k++); k-2;}
a(n) = if(n == 1, 0, A072649(1 + vecmax(factor(n)[,2])));

a(n) = A072649(1 + A051903(n)) for n >= 2.
a(n) = A072649(A375430(n)) + 1 for n >= 2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3 - 1/zeta(2) + Sum_{k>=5} (1 - 1/zeta(Fibonacci(k)-1)) = 2.47666161947309359914... .
If the chosen index for 1 is 1 instead of 2, then the asymptotic mean is 3 - 2/zeta(2) + Sum_{k>=5} (1 - 1/zeta(Fibonacci(k)-1)) = 1.86873451761906697048... .

A375432 Numbers k such that A375428(k) > A375430(k).

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 64, 72, 88, 96, 104, 108, 120, 125, 135, 136, 152, 160, 168, 184, 189, 192, 200, 216, 224, 232, 243, 248, 250, 256, 264, 270, 280, 288, 296, 297, 312, 320, 328, 343, 344, 351, 352, 360, 375, 376, 378, 392, 408, 416, 424, 440, 448, 456

Offset: 1

Amiram Eldar, Aug 15 2024

First differs from A374590 at n = 31.
For numbers k that are not in this sequence A375428(k) = A375430(k).
Numbers k such that A051903(k)+1 is not of the form Fibonacci(m)-1, m >= 3.
The asymptotic density of this sequence is 1 - 1/zeta(2) - Sum_{k>=4} (1/zeta(Fibonacci(k)) - 1/zeta(Fibonacci(k)-1)) = 0.12330053981922224451... .

			8 is a term since A375428(8) = 3 > 2 = A375430(8).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000045, A000071, A051903, A374590, A375428, A375430.

fibQ[n_] := n >= 2 && Or @@ IntegerQ /@ Sqrt[5*n^2 + {-4, 4}]; Select[Range[300], !fibQ[Max[FactorInteger[#][[;;, 2]]] + 1] &]

isfib(n) = n >= 2 && (issquare(5*n^2-4) || issquare(5*n^2+4));
is(n) = n > 1 && !isfib(vecmax(factor(n)[,2]) + 1);

A375428 The maximum exponent in the unique factorization of n in terms of distinct terms of A115975 using the Zeckendorf representation of the exponents in the prime factorization of n; a(1) = 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 2, 2, 1, 1, 1, 3, 1

Offset: 1

Amiram Eldar, Aug 15 2024

Differs from A095691 and A365552 at n = 1, 32, 36, 64, 72, 96, 100, ... . Differs from A368105 at n = 1, 36, 72, 100, 108, ... .

When the exponents in the prime factorization of n are expanded as sums of distinct Fibonacci numbers using the Zeckendorf representation (A014417), we get a unique factorization of n in terms of distinct terms of A115975, i.e., n is represented as a product of prime powers (A246655) whose exponents are Fibonacci numbers. a(n) is the maximum exponent of these prime powers. Thus all the terms are Fibonacci numbers.

			For n = 16 = 2^4, the Zeckendorf representation of 4 is 101, i.e., 4 = Fibonacci(2) + Fibonacci(4) = 1 + 3. Therefore 16 = 2^(1+3) = 2^1 * 2^3, and a(16) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000045, A014417, A051903, A087172, A115975, A246655, A318465, A368781, A375429, A375430.

A087172[n_] := Module[{k = 2}, While[Fibonacci[k] <= n, k++]; Fibonacci[k-1]]; a[n_] := A087172[Max[FactorInteger[n][[;;, 2]]]]; a[1] = 0; Array[a, 100]

A087172(n) = {my(k = 2); while(fibonacci(k) <= n, k++); fibonacci(k-1);}
a(n) = if(n == 1, 0, A087172(vecmax(factor(n)[,2])));

a(n) = A087172(A051903(n)) for n >= 2.
a(n) = A000045(A375429(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 - 1/zeta(2) + Sum_{k>=4} Fibonacci(k) * (1 - 1/zeta(Fibonacci(k))) = 1.64419054900327345836... .

cp's OEIS Frontend

A375431 The indices of the terms of A375430 in the Fibonacci sequence.

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A375432 Numbers k such that A375428(k) > A375430(k).

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A375428 The maximum exponent in the unique factorization of n in terms of distinct terms of A115975 using the Zeckendorf representation of the exponents in the prime factorization of n; a(1) = 0.

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