A375428 -id:A375428 - cp's OEIS frontend

A375429 The indices of the terms of A375428 in the Fibonacci sequence.

0, 2, 2, 3, 2, 2, 2, 4, 3, 2, 2, 3, 2, 2, 2, 4, 2, 3, 2, 3, 2, 2, 2, 4, 3, 2, 4, 3, 2, 2, 2, 5, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 4, 3, 3, 2, 3, 2, 4, 2, 4, 2, 2, 2, 3, 2, 2, 3, 5, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 3, 3, 2, 2, 2, 4, 4, 2, 2, 3, 2, 2, 2, 4, 2, 3, 2, 3, 2, 2, 2, 5, 2, 3, 3, 3, 2, 2, 2, 4, 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 = 16 = 2^4, the Zeckendorf representation of 4 is 101, i.e., 4 = Fibonacci(2) + Fibonacci(4). Therefore 16 = 2^(Fibonacci(2) + Fibonacci(4)) = 2^Fibonacci(2) * 2^Fibonacci(4), and a(16) = 4.

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

Cf. A000045, A051903, A130233, A375428, A375431.

A130233[n_] := Module[{k = 2}, While[Fibonacci[k] <= n, k++]; k-1]; a[n_] := A130233[Max[FactorInteger[n][[;;, 2]]]]; a[1] = 0; Array[a, 100]

A130233(n) = {my(k = 2); while(fibonacci(k) <= n, k++); k-1;}
a(n) = if(n == 1, 0, A130233(vecmax(factor(n)[,2])));

a(n) = A130233(A375428(n)).
a(n) = A130233(A051903(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3 - 1/zeta(2) + Sum_{k>=4} (1 - 1/zeta(Fibonacci(k))) = 2.59996215929231584366... .
If the chosen index for 1 is 1 instead of 2, then the asymptotic mean is 3 - 2/zeta(2) + Sum_{k>=4} (1 - 1/zeta(Fibonacci(k))) = 1.99203505743828921499... .

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);

A375430 The maximum exponent in the unique factorization of n in terms of distinct terms of A115975 using the dual 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, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 2, 2, 2, 1, 1, 1, 2, 1

Offset: 1

Amiram Eldar, Aug 15 2024

First differs from A299090 at n = 128. Differs from A046951 and A159631 at n = 1, 36, 64, 72, ... .

When the exponents in the prime factorization of n are expanded as sums of distinct Fibonacci numbers using the dual Zeckendorf representation (A104326), 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 = 8 = 2^3, the dual Zeckendorf representation of 3 is 11, i.e., 3 = Fibonacci(2) + Fibonacci(3) = 1 + 2. Therefore 8 = 2^(1+2) = 2^1 * 2^2, and a(8) = 2.

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

Cf. A000045, A095791, A104326, A115975, A246655, A331109, A368781, A375428, A375431.

Cf. A046951, A159631, A299090.

A130312[n_] := Module[{k = 0}, While[Fibonacci[k] <= n, k++]; Fibonacci[k-2]]; a[n_] := A130312[1 + Max[FactorInteger[n][[;;, 2]]]]; a[1] = 0; Array[a, 100]

A130312(n) = {my(k = 0); while(fibonacci(k) <= n, k++); fibonacci(k-2);}
a(n) = if(n == 1, 0, A130312(1 + vecmax(factor(n)[,2])));

a(n) = A130312(1 + A051903(n)).
a(n) = A000045(A375431(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=4} Fibonacci(k) * (1 - 1/zeta(Fibonacci(k)-1)) = 1.48543763231328442311... .

A386468 The maximum exponent in the prime factorization of the largest exponentially squarefree divisor of n.

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, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Jul 22 2025

First differs from A375428 at n = 64.
Differs from A368105 at n = 1, 36, 64, 72, 100, ... .
Except for a(1), all the terms are by definition squarefree numbers.

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

Cf. A005117, A051903, A070321, A365683, A368105, A375428, A378085.

a[n_] := Module[{k = Max[FactorInteger[n][[;; , 2]]]}, While[! SquareFreeQ[k], k--]; k]; a[1] = 0; Array[a, 100]

a(n) = if(n == 1, 0, my(k = vecmax(factor(n)[,2])); while(!issquarefree(k), k--); k);

a(n) = A051903(A365683(n)).
a(n) = A070321(A051903(n)) for n >= 2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=2} A378085(k-1)*(1-1/zeta(k)) = 1.66055078443790141429... .

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A375429 The indices of the terms of A375428 in the Fibonacci sequence.

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

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A375430 The maximum exponent in the unique factorization of n in terms of distinct terms of A115975 using the dual Zeckendorf representation of the exponents in the prime factorization of n; a(1) = 0.

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A386468 The maximum exponent in the prime factorization of the largest exponentially squarefree divisor of n.

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