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A377142
Numbers m such that phi(2*m-1)/2 = phi(2*m) - 1, where phi = A000010.
0
2, 4, 5, 16, 64, 4096, 65536, 262144
OFFSET
1,1
COMMENTS
Conjecture 1: each term has the form p^(q-1), where p, q both some primes.
Conjecture 2: sequence is infinite.
Presumably the sequence of numbers of the form (exponent of a(n)) + (smallest divisor of a(n)) is a supersequence of Mersenne exponents.
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
FORMULA
a(n) = (A376337(n) + 1)/2.
EXAMPLE
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;
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.
MAPLE
filter:= m -> numtheory:-phi(2*m-1)/2 = numtheory:-phi(2*m)-1:
select(filter, [$1..10^7]); # Robert Israel, Oct 20 2024
MATHEMATICA
Select[Range[300000], EulerPhi[2*# - 1]/2 == EulerPhi[2*#] - 1 &] (* Amiram Eldar, Oct 30 2024 *)
PROG
(Magma) [m: m in [2..2*10^6] | EulerPhi(2*m-1)/2 eq EulerPhi(2*m)-1];
(PARI) isok(m) = eulerphi(2*m-1)/2 == eulerphi(2*m) - 1; \\ Michel Marcus, Oct 30 2024
CROSSREFS
Supersequence of A019279 and A061652.
Sequence in context: A250545 A070917 A038757 * A062549 A169602 A092051
KEYWORD
nonn,more
AUTHOR
STATUS
approved