A388303 - OEIS

A388303

A classification of the nonnegative integers, where each number lies in exactly one of six classes.

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

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OFFSET

0,5

COMMENTS

Although this is a list, we set the offset to 0 for more intuitive usage: a(n) is the index of the class to which the integer n belongs. There are six mutually exclusive classes, and the nonnegative integers are the union of these classes.

The six classes are defined as follows: n is in class C0 <=> n in {0, 1}. For n >= 2, let F(n) represent the canonical prime factorization of n, p the sequence of prime factors in F(n), and e the corresponding sequence of exponents. Define three Boolean expressions: L = [card(p) = 1], M = [max(e) = 1], and E = [card(set(e)) = 1], where [.] denotes the Iverson bracket. Then n is in C1 <=> L and E and M; n is in C2 <=> L and E and not M; n is in C3 <=> not L and not E and not M; n is in C4 <=> not L and E and M; n is in C5 <=> not L and E and not M. This amounts to the mapping: C0 -> {0, 1}, C1 -> A000040, C2 -> A246547, C3 -> A059404, C4 -> A120944, C5 -> A303606. Finally a(n) = k <=> n in Ck; k = 0..5.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Peter Luschny, Python implementation.

EXAMPLE

|n| |L| |M| |E| |class|

----------------------------------------

32: True False True [C2] -> A246547;

36: False False True [C5] -> A303606;

42: False True True [C4] -> A120944;

43: True True True [C1] -> A000040;

44: False False False [C3] -> A059404.

MATHEMATICA

Table[Which[n < 2, 0, PrimeQ[n], 1, PrimePowerQ[n], 2, SquareFreeQ[n], 4, Length[Union@ FactorInteger[n][[;; , -1]]] == 1, 5, True, 3], {n, 0, 86}] (* Michael De Vlieger, Sep 17 2025 *)

PROG

(Python) # Cf. links.

(SageMath)

def a(n: int) -> int:

if n < 2: return 0

factors = list(factor(n))

exponents = [f[1] for f in factors]

L = 1 == len(factors)

E = 1 == len(set(exponents))

M = 1 == max(exponents)

if L and E: return 1 if M else 2

if E: return 4 if M else 5