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.

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

1,4

Comments

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.

Links

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

Formula

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... .

Example

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.

Mathematica

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]

Prog

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

Crossrefs

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

Sequence in context: A360721 A365552 A095691 * A368105 A182426 A371733

Adjacent sequences: A375425 A375426 A375427 * A375429 A375430 A375431

Keyword

nonn,easy,base

Author

Amiram Eldar, Aug 15 2024

Status

approved