[seqfan] Nonmembers of Range of Euler Phi Function

David Harden oddleehr at alum.mit.edu
Sun Sep 19 19:06:14 CEST 2010

I want to submit a sequence whose nth term is, if it exists, the smallest positive integral multiple of 2^n not in the range of the Euler phi function.
I know the terms up to n=5 (so the offset is n=0): 3, 14, 68, 152, 304, 608

Does anyone know how to prove a_n exists for all n? The other natural question is: Is a_n ever a multiple of 2^(n+1)?

An easy note: If r is in the range of the phi function, so is 2r.
Proof. If phi(k) = r and k is odd, then phi(4k) = 2r. If k is even, then phi(2k) = 2r.
Therefore, if a_n is not a multiple of 2^(n+1), a_(n+1)/a_n >= 2 (assuming a_(n+1) exists).

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