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# Shadow transform

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Given an integer sequence a(0), a(1), ... its shadow transform is the sequence s(0), s(1), ... where s(n) = number of i with 0 <= i < n such that n divides a(i). Note that s(n) <= n.

Halbeisen & Hungerbuehler defined the shadow transform, and proved that that A072453, the shadow transform of A000522, is multiplicative. More generally, any arithmetic function with the reduction property has a shadow transform which is multiplicative, where the reduction property is

${\displaystyle f(n)\equiv f(n\,{\bmod {\,}}q){\pmod {q}}}$ for all ${\displaystyle q\geq 1}$.

There are 1, 1, 1, 3, 12, 48, 288, ... (A226443) shadow transforms on sequences with 0, 1, 2, ... elements.

## Cite this page as

Charles R Greathouse IV, Shadow transform.— From the On-Line Encyclopedia of Integer Sequences® Wiki (OEIS® Wiki). [https://oeis.org/wiki/Shadow_transform]