OEIS A000026: Mosaic Numbers

Great question about the term ‘shadow.’ In mosaic numbers, the mosaic of n is a projection of n’s prime-factor data: it keeps the primes and their exponents, but compresses that information into a single number. Concretely, if n = ∏ p_j^{e_j}, then the mosaic number is mosaic(n) = ∏ (p_j · e_j). This is a kind of shadow because you lose a lot of the original structure—you don’t recover the exact prime bases or the individual exponents from mosaic(n) alone, and different n can map to the same mosaic number. For example: - n = 24 = 2^3 · 3^1 → mosaic(24) = (2·3) · (3·1) = 6 · 3 = 18- n = 6 = 2^1 · 3^1 → mosaic(6) = (2·1) · (3·1) = 2 · 3 = 6- n = 8 = 2^3 → mosaic(8) = (2·3) = 6So both 6 and 8 share mosaic(…) = 6, illustrating the non-invertible, shadow-like nature of the map. For square-free numbers, where every exponent e_j = 1, mosaic(n) = ∏ p_j = n, so in that special case the mosaic number matches the original number exactly. This “shadow” view helps explain why mosaic numbers reveal only a facet of the factorization, yet connect to other OEIS ideas—like how the largest prime factor and the total sum of exponents relate to mosaic numbers—and why the mosaic map interacts with divisors and average orders in interesting ways. We’ll explore those connections and what they tell us about the structure beneath the surface.

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