slug: oeis-a388986-euler-form-divisor-sum bibkey: oeis2026a388986 doi: null url: https://oeis.org/A388986 triage: theorem motivation_gids:

• D5/S3/Arith/EulerFormDivisorSum

The A388986 Euler-form inclusion question

Problem

Does every number in A228058 belong to A388986? Explicitly, let p be prime, p congruent to 1 modulo 4, a a natural number, and r an odd integer greater than one with gcd(p,r)=1. Put N=p^(4a+1)r^2. The question asks whether the sum of the unitary divisors of N plus the sum of its squarefree divisors is strictly less than 2N.

Motivation

This is a first-tier recent OEIS question. A388986 revision 36, read on September 9, 2026, still asks whether A228058 is a subsequence. The Euler form describes candidates for odd perfect numbers, but this question does not assert the existence or nonexistence of odd perfect numbers.

Gap

The complete main entry and its 15 direct A references contain no proof of the inclusion. A389079 and A389219 repeat it as a conjecture. The known prime-product formulas for each divisor sum do not alone supply the joint strict estimate. Finch’s Unitarism and Infinitarism and Trudgian’s The sum of the unitary divisor function were also checked; their results concern parametrizations and asymptotic estimates, not this Euler-form inequality.

Route

Parametrize a unitary divisor by its prime support: coprimality forces its exponents to be the full exponents in N. Injecting these supports into the powerset bounds the unitary sum by the product of q^e+1. Mathlib’s squarefree divisor powerset identity gives the other sum as the product of q+1.

Divide by N>0. A finite minimum-induction argument bounds the product of 1+1/q^2 over integers q>=b>1 by b/(b-1). For the nonempty prime support of r, split into no 3, only 3, and 3 with another prime. These cases bound the sum of the two normalized support products strictly below 5/3. The distinguished prime p>=5 contributes at most 6/5 to either product, giving the strict bound below 2. Larger exponents only decrease the normalized factors.

Falsifier

An admissible triple p,a,r whose actual two divisor sums add to at least 2N would contradict the formal statement. The excluded case p=5,a=0,r=1 gives N=5 and 6+6>10, so nonempty remaining support is essential.

Evidence

The Lean theorem is D5/S3/Arith/EulerFormDivisorSum.euler_form_lt. The definitions unitarySum and squarefreeSum use filtered Nat.divisors directly. There is no finite cutoff in the statement or proof.

The continuation worker independently checked all positive N through 10^6: both divisor-sieve sums agree with the prime-product formulas everywhere; all 12,941 Euler-form instances pass. At p=5,a=0,r=3, N=45 and 60+24<90. The largest observed ratio is 28/15, attained at 45. This enumeration is a diagnostic check and is not used by the Lean proof.

Triage

Theorem: the exact A228058 inclusion question is answered affirmatively. The separate 2026 prime-factorization-subset conjecture on the same OEIS page is outside this result.

ASSUMED-UNVERIFIED

Publication priority and absence of unindexed proofs are not established. The source-to-formal identification is documentary; the Lean kernel checks the explicit filtered sums and all quantified hypotheses. The continuation worker rechecked the inherited proofs and numerical claims; it does not claim independent multi-model review or verification by the caller.