nonmultiplicativeby4

nonmultiplicativeby4 is a term used in number theory to describe integers that are not divisible by 4. In terms of prime factorisation, a nonmultiplicativeby4 integer has an exponent of the prime 2 that is less than two; that is, the 2‑adic valuation v₂(n) is either 0 or 1. Consequently the set of nonmultiplicativeby4 numbers consists of all odd integers together with all integers of the form 2k where k is an odd integer. Examples of nonmultiplicativeby4 numbers are 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, and so on. The complement of this set—numbers that are divisible by 4—has v₂(n) at least 2.

The classification of numbers into nonmultiplicativeby4 and divisible-by-4 is useful in a range of mathematical contexts. For instance, the parity of the sum of two integers depends on whether each integer is nonmultiplicativeby4 or divisible by 4. The property also arises in studies of quadratic residues, where the residue of a nonmultiplicativeby4 number modulo 8 determines whether it is a quadratic residue mod 8. In algorithmic number theory, tests for divisibility by 4 often use the last two binary digits of a number, exploiting the fact that a nonmultiplicativeby4 integer ends with 01, 10, or 11 in binary.

In the OEIS, the sequence of nonmultiplicativeby4 numbers appears under A004000, identified as “Numbers n such that n is not divisible by 4.” The sequence is used as a reference point for generating or filtering numbers according to their divisibility by powers of 2, which is a common requirement in cryptographic primitives, integer sequence analysis, and computational number theory.