The OEIS A103585 Sloping-Binary Period Conjecture

Abstract

The 129th and 172nd entries refute the proposed period 43 of OEIS A103585.

Definition 1.1 (The sloping-binary sequence).

$$\forall k\in \mathrm{Nat},s\left(k\right)=k+\sum _{m\in \mathrm{Icc}\left(1,k+1\right)}(\mathrm{ite}\left(\left(k+m\right)\mathrm{mod}{2}^m=0,2^m,0\right))$$

Formalization. D5/S0/Certificates/StephanSlopingBinaryPeriodRefutation.s (✓ std3).

Citation. Benoit Cloitre; Philippe Deléham; Ralf Stephan (2005). OEIS A103585, numbers k with (A102370(k)-k)/2 = 1, read mod 4. URL: https://oeis.org/A103585.

Commentary.

The value s(k) is A102370(k), written as the finite conditional sum in its OEIS formula.

Definition 1.2 (The A103585 selection predicate).

$$\forall k\in \mathrm{Nat},(P\left(k\right))\iff (s\left(k\right)=k+2)$$

Formalization. D5/S0/Certificates/StephanSlopingBinaryPeriodRefutation.P (✓ std3).

Citation. Benoit Cloitre; Philippe Deléham; Ralf Stephan (2005). OEIS A103585, numbers k with (A102370(k)-k)/2 = 1, read mod 4. URL: https://oeis.org/A103585.

Commentary.

The predicate P selects exactly those natural k for which s(k) equals k+2.

Definition 1.3 (The zero-based A103585 sequence).

$$\forall i\in \mathrm{Nat},A\left(i\right)=\mathrm{nth}\left(P,i\right)\mathrm{mod}4$$

Formalization. D5/S0/Certificates/StephanSlopingBinaryPeriodRefutation.A (✓ std3).

Citation. Benoit Cloitre; Philippe Deléham; Ralf Stephan (2005). OEIS A103585, numbers k with (A102370(k)-k)/2 = 1, read mod 4. URL: https://oeis.org/A103585.

Commentary.

The value A(i) is the i-th natural satisfying P, reduced modulo four. Thus OEIS a(m) equals A(m-1).

Definition 1.4 (Stephan’s period-43 claim).

$$(claim)\iff (\forall i\in \mathrm{Nat},A\left(i+43\right)=A\left(i\right))$$

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

Citation. Benoit Cloitre; Philippe Deléham; Ralf Stephan (2005). OEIS A103585, numbers k with (A102370(k)-k)/2 = 1, read mod 4. URL: https://oeis.org/A103585.

Commentary.

The claim asserts that shifting any zero-based index by 43 preserves A.

Theorem 1.5 (The period-43 claim fails).

$$\neg claim$$

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

Resolves. Problems/oeis-a103585-sloping-binary-period-refutation (refuted) by D5/S0/Certificates/StephanSlopingBinaryPeriodRefutation.result.

Source. Repository-derived.

Acknowledgement. Benoit Cloitre; Philippe Deléham; Ralf Stephan (2005). OEIS A103585, numbers k with (A102370(k)-k)/2 = 1, read mod 4. URL: https://oeis.org/A103585.

Commentary.

The finite ranking certificate identifies the 129th entry with 383 modulo four and the 172nd entry with 513 modulo four. Hence a(129)=3 differs from a(172)=1. This refutes the period-43 claim without changing the definition of A102370 or asserting a true period.

References

• Truth anchor: D5/S0/Certificates/StephanSlopingBinaryPeriodRefutation.A
• Truth anchor: D5/S0/Certificates/StephanSlopingBinaryPeriodRefutation.P
• Truth anchor: D5/S0/Certificates/StephanSlopingBinaryPeriodRefutation.claim
• Truth anchor: D5/S0/Certificates/StephanSlopingBinaryPeriodRefutation.result
• Truth anchor: D5/S0/Certificates/StephanSlopingBinaryPeriodRefutation.s