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%I #32 Sep 08 2022 08:46:14
%S 3,5,17,257,65537,4294967297
%N Numbers n such that phi(n-2) = phi(n-1) = (n-1) / 2.
%C No more terms below 10^8; 4294967297 is a term of this sequence.
%C First 5 terms are Fermat primes (A019434).
%C Conjecture: next term is 4294967297.
%C Subsequence of A232720 and A000051.
%C Sequence is different from A232720 and A000215; A232720(6) = 83623937 and A000215(7) = 18446744073709551617 are not terms of this sequence.
%C From _Jeppe Stig Nielsen_, Nov 19 2016: (Start)
%C Since n-1 is a solution to phi(x)=x/2, it is clear from the formula for phi that x=n-1 is a nontrivial power of two (in A000079 and even). Then y=n-2 is an odd number such that phi(y) is a power of two, and again recalling the formula for phi, this can only happen when y is a product of distinct Fermat primes (y in A045544).
%C Question: Is the above comment, together with Euler's demonstration that the Fermat number 2^32 + 1 = A000215(5) is composite, enough to prove that A262534 has only these six terms?
%C With the above observations, we need only search next to powers of two (A000051), so it is quick to determine that there are no terms between 65537 and 4294967297. (End)
%C From _Robert Israel_, Dec 08 2016: (Start)
%C The j-th binary digit (i.e. coefficient of 2^j) of the product of a set of distinct Fermat numbers, say y=Product_{k in T} (2^(2^k)+1), is 1 iff j = Sum_{k in S} 2^k for some subset S of T. In order for x = y+1 to be a power of 2, all of y's binary digits must be 1. Since 2^(2^5)+1 is composite, T cannot contain 5, so digit 32 is 0, and y < 2^32. Thus we do have only these 6 terms. (End)
%e 17 is in this sequence because phi(15) = phi(16) = 8 = (17 - 1) / 2.
%t Select[Range@ 100000, EulerPhi[# - 2] == EulerPhi[# - 1] == (# - 1)/2 &] (* _Michael De Vlieger_, Sep 25 2015 *)
%o (Magma) [n: n in [3..10000000] | n-1 eq 2*EulerPhi(n-1) and n-1 eq 2*EulerPhi(n-2)]
%o (PARI) for(n=1, 1e8, if(eulerphi(n-2) == eulerphi(n-1) && 2*eulerphi(n-1) == (n-1), print1(n ", "))) \\ _Altug Alkan_, Oct 11 2015
%Y Cf. A000010, A000215, A232720, A000079, A045544, A000051, A019434.
%K nonn,fini,full
%O 1,1
%A _Jaroslav Krizek_, Sep 24 2015
%E a(6) from _Jeppe Stig Nielsen_, Nov 19 2016
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