cp's OEIS Frontend

A377142 Numbers m such that phi(2*m-1)/2 = phi(2*m) - 1, where phi = A000010.

%I A377142 #41 Nov 17 2024 07:38:02
%S A377142 2,4,5,16,64,4096,65536,262144
%N A377142 Numbers m such that phi(2*m-1)/2 = phi(2*m) - 1, where phi = A000010.
%C A377142 Conjecture 1: each term has the form p^(q-1), where p, q both some primes.
%C A377142 Conjecture 2: sequence is infinite.
%C A377142 Presumably the sequence of numbers of the form (exponent of a(n)) + (smallest divisor of a(n)) is a supersequence of Mersenne exponents.
%C A377142 If 2*m-1 is a Mersenne prime (A000668), then phi(2*m-1)/2 = m-1 = phi(2*m) - 1, so m is a term. - _Robert Israel_, Oct 20 2024
%F A377142 a(n) = (A376337(n) + 1)/2.
%e A377142 2 is a term because phi(2*2-1)/2 = phi(3)/2 = 2/2 = 1 is equal to phi(2*2)-1 = phi(4)-1 = 2-1 = 1;
%e A377142 5 is a term because phi(2*5-1)/2 = phi(9)/2 = 6/2 = 3 is equal to phi(2*5)-1 = phi(10)-1 = 4-1 = 3.
%p A377142 filter:= m -> numtheory:-phi(2*m-1)/2 = numtheory:-phi(2*m)-1:
%p A377142 select(filter, [$1..10^7]); # _Robert Israel_, Oct 20 2024
%t A377142 Select[Range[300000], EulerPhi[2*# - 1]/2 == EulerPhi[2*#] - 1 &] (* _Amiram Eldar_, Oct 30 2024 *)
%o A377142 (Magma) [m: m in [2..2*10^6] | EulerPhi(2*m-1)/2 eq EulerPhi(2*m)-1];
%o A377142 (PARI) isok(m) = eulerphi(2*m-1)/2 == eulerphi(2*m) - 1; \\ _Michel Marcus_, Oct 30 2024
%Y A377142 Supersequence of A019279 and A061652.
%Y A377142 Cf. A000010, A000043, A000668, A090748, A291901, A376337.
%K A377142 nonn,more
%O A377142 1,1
%A A377142 _Juri-Stepan Gerasimov_, Oct 19 2024