A250405 -id:A250405 - cp's OEIS frontend

A250404 Numbers k such that the set of all distinct values of phi of all divisors of k equals the set of all proper divisors of k+1 where phi is the Euler totient function (A000010).

1, 2, 3, 15, 255, 65535, 4294967295

Offset: 1

Jaroslav Krizek, Nov 22 2014

Numbers k such that {phi(d) : d|k} = {d : d|(k+1), d<(k+1)} as sets.
Conjecture: last term is 4294967295.
Sequence differs from A203966 because 83623935 is not in this sequence.

			2 is a term since {phi(d) : d|2} = {1} = {d; d|2, d<2}.
15 is a term since {phi(d) : d|15} = {1, 2, 4, 8} = {d : d|16, d<16}.

Subsequence of A203966.

Cf. A000010, A203966, A250405.

[n: n in [1..100000] | Set([EulerPhi(d): d in Divisors(n)]) eq Set([d: d in Divisors(n+1) | d lt n+1 ])]

isok(n) = {sphi = []; fordiv(n, d, sphi = Set(concat(sphi, eulerphi(d)))); sphi == setminus(Set(divisors(n+1)), Set(n+1));} \\ Michel Marcus, Nov 23 2014

Edited and a(7) added by Max Alekseyev, May 04 2024

A382519 Odd positive integers m such that phi(m) and phi(m+1) are both powers of 2.

Original entry on oeis.org

1, 3, 5, 15, 255, 65535, 4294967295

Offset: 1

Caleb Stanford, Apr 05 2025

Sequence is finite with only 7 values. With the exception of m = 5, the others are products of the first k Fermat primes; i.e., products of A019434 and matching the initial terms of A051179. With the exception of m = 5, sequence resembles A250405.

			5 is present because phi(5) = 4 and phi(6) = 2, both powers of two.
15 is present because phi(15) = 8 and phi(16) = 8, both powers of two.
17 is not present because phi(17) = 16 but phi(18) = 6, not a power of two.

John and Caleb Stanford (Math StackExchange), A possible Property of Euler's totient function: n such that phi(n) and phi(n+1) are both powers of two

Cf. A051179, A000215, A019434, A003401, A250405.

Subsequence of A382803.

a(n) = 2^2^k - 1 for k = 0, 1, 2, 3, 4, 5, equivalently the product of first k Fermat numbers, OR a(n) = 5. Sequence is finite because the next Fermat number, 4294967297 is composite.

cp's OEIS Frontend

A250404 Numbers k such that the set of all distinct values of phi of all divisors of k equals the set of all proper divisors of k+1 where phi is the Euler totient function (A000010).

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