The OEIS A245786 Subsequence Conjecture

Abstract

The seventh listed A245786 member is not an A216793 gcd record.

Definition 1.1 (Membership in A245786).

$$\forall n\in \mathbb{N},(IsMember\left(n\right))\iff (\exists z\in \mathbb{Z},\frac{(n:\mathbb{Q})}{(\left({\sigma }_0\right)\left(n\right):\mathbb{Q})}+\frac{(\left({\sigma }_1\right)\left(n\right):\mathbb{Q})}{(n:\mathbb{Q})}=z)$$

Formalization. D5/S0/Certificates/KrizekIntegralityGcdRecordRefutation.IsMember (✓ std3).

Citation. Jaroslav Krizek (2014). OEIS A245786, Numbers n such that k(n) = (n/tau(n) + sigma(n)/n) is an integer. URL: https://oeis.org/A245786.

Commentary.

Here sigma sub zero is tau, the divisor-count function, and sigma sub one is the divisor-sum function. Membership means that the displayed rational sum is an integer.

Definition 1.2 (The A216793 gcd record predicate).

$$\forall n\in \mathbb{N},(IsRecord\left(n\right))\iff (\forall m\in \mathbb{N},0<m\Rightarrow \left(m<n\Rightarrow \gcd \left(\left({\sigma }_1\right)\left(m\right),m\right)<\gcd \left(\left({\sigma }_1\right)\left(n\right),n\right)\right))$$

Formalization. D5/S0/Certificates/KrizekIntegralityGcdRecordRefutation.IsRecord (✓ std3).

Citation. Jaroslav Krizek (2014). OEIS A245786, Numbers n such that k(n) = (n/tau(n) + sigma(n)/n) is an integer. URL: https://oeis.org/A245786.

Commentary.

At a natural n, the gcd of sigma sub one of n with n must be strictly larger than the corresponding gcd at every positive smaller m.

Definition 1.3 (Krizek’s subsequence conjecture).

$$(claim)\iff (\forall n\in \mathbb{N},0<n\Rightarrow \left(IsMember\left(n\right)\Rightarrow IsRecord\left(n\right)\right))$$

Formalization. D5/S0/Certificates/KrizekIntegralityGcdRecordRefutation.claim (✓ std3).

Citation. Jaroslav Krizek (2014). OEIS A245786, Numbers n such that k(n) = (n/tau(n) + sigma(n)/n) is an integer. URL: https://oeis.org/A245786.

Commentary.

The conjecture says that every positive A245786 member is an A216793 gcd record.

Theorem 1.4 (The conjecture fails at n = 275890944).

$$\neg claim$$

Proof. Machine-checked in Lean as D5/S0/Certificates/KrizekIntegralityGcdRecordRefutation.result (✓ std3). ∎

Resolves. Problems/oeis-a245786-integrality-gcd-record-refutation (refuted) by D5/S0/Certificates/KrizekIntegralityGcdRecordRefutation.result.

Source. Repository-derived.

Acknowledgement. Jaroslav Krizek (2014). OEIS A245786, Numbers n such that k(n) = (n/tau(n) + sigma(n)/n) is an integer. URL: https://oeis.org/A245786.

Commentary.

At N = 275890944, tau(N) = 288 and sigma(N) = 919636480, so N/tau(N) + sigma(N)/N = 957958. Its gcd value is 91963648. For the smaller M = 142990848, sigma(M) = 571963392 = 4M and the gcd value is M itself. Thus N is a member but not a record.

References

• Truth anchor: D5/S0/Certificates/KrizekIntegralityGcdRecordRefutation.IsMember
• Truth anchor: D5/S0/Certificates/KrizekIntegralityGcdRecordRefutation.IsRecord
• Truth anchor: D5/S0/Certificates/KrizekIntegralityGcdRecordRefutation.claim
• Truth anchor: D5/S0/Certificates/KrizekIntegralityGcdRecordRefutation.result