A382519 -id:A382519 - cp's OEIS frontend

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A382803 Positive integers m such that phi(m) and phi(m+1) are both powers of 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 15, 16, 255, 256, 65535, 65536, 4294967295

Offset: 1

Caleb Stanford, Apr 05 2025

Numbers m such that m and m+1 are in A003401
Each of m and m+1 must be a power of 2 times a product of Fermat primes.
Apart from term 5, odd terms are of the form 2^2^k - 1 for k in 0...5.
Even terms are exactly numbers of the form 2^2^k such that 2^2^k + 1 is a Fermat prime (A019434).
The sequence is thus infinite iff A019434 is infinite, i.e., iff there are infinitely many Fermat primes.

			16 is present because phi(16) = 8 and phi(17) = 16, both powers of two.
17 is not present because phi(17) = 16 but phi(18) = 6, not a power of two.

User John and Caleb Stanford (Math StackExchange), A possible Property of Euler's totient function: n such that phi(n) and phi(n+1) are both powers of two

Cf. A000010, A000215, A003401, A019434, A051179, A382519 (odd terms).

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