A291901 -id:A291901 - cp's OEIS frontend

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

2, 4, 5, 16, 64, 4096, 65536, 262144

Offset: 1

Juri-Stepan Gerasimov, Oct 19 2024

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

			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.

Supersequence of A019279 and A061652.

Cf. A000010, A000043, A000668, A090748, A291901, A376337.

[m: m in [2..2*10^6] | EulerPhi(2*m-1)/2 eq EulerPhi(2*m)-1];

filter:= m -> numtheory:-phi(2*m-1)/2 = numtheory:-phi(2*m)-1:
select(filter, [$1..10^7]); # Robert Israel, Oct 20 2024

Select[Range[300000], EulerPhi[2*# - 1]/2 == EulerPhi[2*#] - 1 &] (* Amiram Eldar, Oct 30 2024 *)

isok(m) = eulerphi(2*m-1)/2 == eulerphi(2*m) - 1; \\ Michel Marcus, Oct 30 2024

a(n) = (A376337(n) + 1)/2.

cp's OEIS Frontend

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

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

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