Erythema Toxicum Neonatorum (ETN): This is a common, benign condition that appears as a fine papular rash, often on the torso, in newborns. It typically presents within the first few days of life but can persist or appear up to 2 weeks of age, making it a plausible diagnosis in a 6-week-old infant who may have had a delayed onset or prolonged course.

Atopic Dermatitis (Eczema): Although it more commonly starts after 3 months, atopic dermatitis can begin earlier, presenting with a papular rash on the torso among other areas. Family history of atopy and associated symptoms like itching (if the infant is old enough to demonstrate) can support this diagnosis.
Seborrhoeic Dermatitis: This condition can cause a fine papular rash and is common in infants, often presenting on the scalp (cradle cap) but can also affect the torso. It tends to appear in the first few months of life.
Contact Dermatitis: This is a reaction to something that has come into contact with the skin, such as a detergent or lotion. It can present as a fine papular rash on the areas of the body that came into contact with the irritant.

Congenital Syphilis: Although less common, congenital syphilis can present with a rash among other systemic symptoms. The rash can be papular and is a critical diagnosis not to miss due to the severe consequences of untreated syphilis.
Infantile Acropustulosis: A rare condition but one that can present with a papular rash, often on the hands and feet but can spread to other areas. It's more common in infants of African descent.
Scabies: Caused by the mite Sarcoptes scabiei, scabies can present with a papular rash and is highly contagious. In infants, the rash can appear on the palms, soles, and face, in addition to the torso.

Incontinentia Pigmenti: A genetic disorder that can cause a rash that evolves through several stages, including a papular stage. It's rare and often associated with other systemic and developmental abnormalities.
Langerhans Cell Histiocytosis: A rare disorder in which the body accumulates too many immature Langerhans cells, leading to a variety of symptoms including skin rashes. The rash can be papular and is often accompanied by other systemic symptoms.
Leiner Disease: Also known as severe combined immunodeficiency with erythroderma, it presents with a severe rash and immunodeficiency. It's extremely rare and would be considered in infants with significant immune compromise and a rash that doesn't respond to typical treatments.