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A382519 Odd positive integers m such that phi(m) and phi(m+1) are both powers of 2.

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%I A382519 #37 Apr 19 2025 06:18:34
%S A382519 1,3,5,15,255,65535,4294967295
%N A382519 Odd positive integers m such that phi(m) and phi(m+1) are both powers of 2.
%C A382519 Sequence is finite with only 7 values. With the exception of m = 5, the others are products of the first k Fermat primes; i.e., products of A019434 and matching the initial terms of A051179. With the exception of m = 5, sequence resembles A250405.
%H A382519 John and Caleb Stanford (Math StackExchange), <a href="https://math.stackexchange.com/questions/1886835">A possible Property of Euler's totient function: n such that phi(n) and phi(n+1) are both powers of two</a>
%F A382519 a(n) = 2^2^k - 1 for k = 0, 1, 2, 3, 4, 5, equivalently the product of first k Fermat numbers, OR a(n) = 5. Sequence is finite because the next Fermat number, 4294967297 is composite.
%e A382519 5 is present because phi(5) = 4 and phi(6) = 2, both powers of two.
%e A382519 15 is present because phi(15) = 8 and phi(16) = 8, both powers of two.
%e A382519 17 is not present because phi(17) = 16 but phi(18) = 6, not a power of two.
%Y A382519 Cf. A051179, A000215, A019434, A003401, A250405.
%Y A382519 Subsequence of A382803.
%K A382519 nonn,fini,full
%O A382519 1,2
%A A382519 _Caleb Stanford_, Apr 05 2025