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

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

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

References

• Lorenz Halbeisen and Norbert Hungerbuehler, "Number theoretic aspects of a combinatorial function," Notes on Number Theory and Discrete Mathematics 5 (1999), pp. 138-150.
• Lorenz Halbeisen, A number-theoretic conjecture and its implication for set theory, Acta Math. Univ. Comenianae 74:2 (2005), pp. 243-254.

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]