bibkey: oeis2005a108569 authors: OEIS contributors year: 2005 title: “OEIS A108569, numbers k with phi(k) = phi(k + phi(k))” doi: null url: https://oeis.org/A108569 claim: “The sequence lists the positive integers k with phi(k) = phi(k + phi(k)). Conjecture: except for the first term, all terms are even; equivalently, no odd k > 1 satisfies the equation.” strata_touched: [] license: citation-only triage: anchor
OEIS A108569, fixed points of phi under the shift by phi
The parity conjecture quoted above is open and nothing in this repository proves it. An attempt produced no theorem. Its value is the set of handles it closed, recorded here with the smallest case that closes each.
The terms below 40000 are 1, 4, 8, 16, 32, 64, 110, 128, 220, 256, 440, 506, 512, 550, ...,
ninety-four in all, and after the first there is no odd one.
For odd k > 1, phi(k) is even and less than k, so m = k + phi(k) is again odd with
k < m < 2k, and the claim is that phi(m) = phi(k) is then impossible.
Handles that do not work
| handle | first failure | values there |
|---|---|---|
| an inequality between the odd parts | k = 3 | phi(k) = 2, m = 5, phi(m) = 4; both odd parts are 1 |
| an inequality between the 2-adic valuations | k = 13 | phi(k) = 12, m = 25, phi(m) = 20; both valuations are 2 |
assuming gcd(k, m) = 1 | k = 9 | phi(k) = 6, m = 15, phi(m) = 8, gcd(k,m) = 3 |
requiring q - 1 to divide phi(k) for every prime q dividing m | k = 5 | phi(k) = 4, m = 9, phi(m) = 6 |
strengthening that to phi(q^a) dividing phi(k) for every exact prime power q^a of m | k = 39 | phi(9) = phi(7) = 6 both divide 24, but their product does not |
| adding equal 2-adic valuations to the prime-power divisibility test | k = 155 | phi(k) = 120, m = 275 = 25 * 11, phi(m) = 200; phi(25) = 20 and phi(11) = 10 both divide 120 |
Local divisibility conditions on the prime powers of m are therefore not enough on their own,
and neither is any of the separations above. A later attempt should start from the k = 155
configuration rather than from the beginning.
Object status
Pinned Mathlib has Mathlib/Data/Nat/Totient.lean; a search of Mathlib/NumberTheory/ for
totient fixed points or totient iteration returned nothing. This repository’s totient material
is D5/S3/Factorization/TotientNondivisorRecords, specific numeric records, and
D5/S3/Factorization/LehmerTotientDichotomy, prime divisor behaviour for the Lehmer problem;
neither concerns fixed points of k mapsto k + phi(k). The OEIS entry carries no comment,
formula or link asserting a proof, read in full on 2026-09-13.
Verified locator
- URL: https://oeis.org/A108569