A375768 -id:A375768 - cp's OEIS frontend

A375769 The indices of the terms of A375768 in the Fibonacci sequence.

0, 2, 2, 3, 2, 2, 2, 4, 3, 2, 2, 3, 2, 2, 2, 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, 3, 3, 2, 3, 2, 4, 2, 4, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 3, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 4, 2, 3, 2, 3, 2, 2, 2, 5, 2, 3, 3, 3, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2

Offset: 1

Amiram Eldar, Aug 27 2024

First differs from A375767 at n = 2448.

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

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

Cf. A000045, A130233, A369939, A375767, A375768.

fibQ[n_] := Or @@ IntegerQ /@ Sqrt[5*n^2 + {-4, 4}]; A130233[n_] := Module[{k = 2}, While[Fibonacci[k] <= n, k++]; k-1]; s[n_] := Module[{e = Max[FactorInteger[n][[;; , 2]]]}, If[fibQ[e], A130233[e], Nothing]]; s[1] = 0; Array[s, 100]

isfib(n) = issquare(5*n^2 - 4) || issquare(5*n^2 + 4);
A130233(n) = {my(k = 2); while(fibonacci(k) <= n, k++); k-1;}
lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = vecmax(factor(k)[,2]); if(isfib(e), print1(A130233(e), ", ")));}

a(n) = A130233(A375768(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (2/zeta(2) + Sum_{k>=3} k * (1/zeta(Fibonacci(k)+1) - 1/zeta(Fibonacci(k)))) / d = 2.4999593748274972257073..., where d = 1/zeta(4) + Sum_{k>=5} (1/zeta(Fibonacci(k)+1) - 1/zeta(Fibonacci(k))) = 0.94462177878047854647... is the density of A369939.
If the chosen index for 1 is 1 instead of 2, then the asymptotic mean is (1/zeta(2) + Sum_{k>=3} k * (1/zeta(Fibonacci(k)+1) - 1/zeta(Fibonacci(k)))) / d = 1.85639269500896710302009... .

A382422 The product of exponents in the prime factorization of the biquadratefree numbers.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1

Offset: 1

Amiram Eldar, Mar 25 2025

Differs from A375766 and A375768 at n = 1, 31, 34, 35, 38, 39, ... .

All the terms are 3-smooth numbers (A003586).

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

Cf. A003586, A005361, A013662, A046100, A082695, A382423, A382424.

Cf. A375766, A375768.

s[n_] := Times @@ FactorInteger[n][[;; , 2]]; biqFreeQ[n_] := Max[FactorInteger[n][[;; , 2]]] < 4; s /@ Select[Range[100], biqFreeQ]

list(kmax) = {my(e); print1(1, ", "); for(k = 2, kmax, e = factor(k)[, 2]; if(vecmax(e) < 4, print1(vecprod(e), ", "))); }

a(n) = A005361(A046100(n)).
a(n) = 2^A382423(n) * 3^A382424(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(4) * Product_{p prime} (1 + 1/p^2 + 1/p^3 - 3/p^4) = 1.57226906210272200398... .
In general, the asymptotic mean of the product of exponents in the prime factorization of the k-free numbers (numbers that are not divisible by a k-th power other than 1), for k >= 2, is zeta(k) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + ... + 1/p^(k-1) - (k-1)/p^k). For k = 2 (squarefree numbers), the mean is 1 since the sequence contains only 1's. The limit when k->oo is zeta(2)*zeta(3)/zeta(6) (A082695).

A375766 The maximum exponent in the prime factorization of the numbers whose exponents in their prime factorization are all Fibonacci numbers.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Aug 27 2024

First differs from A375768 at n = 2448.

All the terms are Fibonacci numbers by definition.

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

Cf. A000045, A051903, A115063, A375274, A375767, A375768.

fibQ[n_] := Or @@ IntegerQ /@ Sqrt[5*n^2 + {-4, 4}]; s[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, fibQ], Max[e], Nothing]]; s[1] = 0; Array[s, 100]

isfib(n) = issquare(5*n^2 - 4) || issquare(5*n^2 + 4);
lista(kmax) = {my(e, ans); print1(0, ", "); for(k = 2, kmax, e = factor(k)[,2]; ans = 1; for(i = 1, #e, if(!isfib(e[i]), ans = 0; break)); if(ans, print1(vecmax(e), ", ")));}

a(n) = A051903(A115063(n)).
a(n) = A000045(A375767(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/zeta(2) + Sum_{k>=3} (Fibonacci(k) * (d(k) - d(k-1)))) / A375274 = 1.52546070254904121983..., where d(k) = Product_{p prime} ((1-1/p)*(1 + Sum_{i=2..k} 1/p^Fibonacci(i))) for k >= 3, and d(2) = 1/zeta(2).

cp's OEIS Frontend

A375769 The indices of the terms of A375768 in the Fibonacci sequence.

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A382422 The product of exponents in the prime factorization of the biquadratefree numbers.

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A375766 The maximum exponent in the prime factorization of the numbers whose exponents in their prime factorization are all Fibonacci numbers.

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