cp's OEIS Frontend

A262534 Numbers k such that phi(k-2) = phi(k-1) = (k-1) / 2.

%I A262534 #35 Jun 22 2025 21:31:55
%S A262534 3,5,17,257,65537,4294967297
%N A262534 Numbers k such that phi(k-2) = phi(k-1) = (k-1) / 2.
%C A262534 No more terms below 10^8; 4294967297 is a term of this sequence.
%C A262534 First 5 terms are Fermat primes (A019434).
%C A262534 Conjecture: next term is 4294967297.
%C A262534 Subsequence of A232720 and A000051.
%C A262534 Sequence is different from A232720 and A000215; A232720(6) = 83623937 and A000215(7) = 18446744073709551617 are not terms of this sequence.
%C A262534 From _Jeppe Stig Nielsen_, Nov 19 2016: (Start)
%C A262534 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 A262534 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 A262534 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 A262534 From _Robert Israel_, Dec 08 2016: (Start)
%C A262534 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 A262534 17 is in this sequence because phi(15) = phi(16) = 8 = (17 - 1) / 2.
%t A262534 Select[Range@ 100000, EulerPhi[# - 2] == EulerPhi[# - 1] == (# - 1)/2 &] (* _Michael De Vlieger_, Sep 25 2015 *)
%o A262534 (Magma) [n: n in [3..10000000] | n-1 eq 2*EulerPhi(n-1) and n-1 eq 2*EulerPhi(n-2)];
%o A262534 (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 A262534 Cf. A000010, A000215, A232720, A000079, A045544, A000051, A019434.
%K A262534 nonn,fini,full
%O A262534 1,1
%A A262534 _Jaroslav Krizek_, Sep 24 2015
%E A262534 a(6) from _Jeppe Stig Nielsen_, Nov 19 2016