The OEIS A129598 Differing-Position Conjecture

Abstract

The value 60 refutes Karttunen’s A089966 characterization of A129598 differences.

Definition 1.1 (The gcd-totient map).

$$\forall m\in \mathrm{Nat},g\left(m\right)=\gcd \left(m,\mathrm{totient}\left(m\right)\right)$$

Formalization. D5/S0/Certificates/KarttunenGcdTotientPositionRefutation.g (✓ std3).

Citation. Antti Karttunen (2007). OEIS A129598, a(n) = n * A111089(n). URL: https://oeis.org/A129598.

Commentary.

For each natural m, g(m) is the greatest common divisor of m and Euler’s totient of m.

Definition 1.2 (The greatest-prime-factor function).

$$\forall n\in \mathrm{Nat},\mathrm{greatestPrimeFactor}\left(n\right)=\mathrm{getLastD}\left(\mathrm{primeFactorsList}\left(n\right),0\right)$$

Formalization. D5/S0/Certificates/KarttunenGcdTotientPositionRefutation.greatestPrimeFactor (✓ std3).

Citation. Antti Karttunen (2007). OEIS A129598, a(n) = n * A111089(n). URL: https://oeis.org/A129598.

Commentary.

The prime factors are listed with multiplicity in increasing order. The last entry is the greatest prime factor, with default zero when the list is empty.

Definition 1.3 (The A129598 sequence).

$$\forall n\in \mathrm{Nat},a\left(n\right)=\mathrm{if}\left(n=1,2,n\cdot \mathrm{greatestPrimeFactor}\left(n\right)\right)$$

Formalization. D5/S0/Certificates/KarttunenGcdTotientPositionRefutation.a (✓ std3).

Citation. Antti Karttunen (2007). OEIS A129598, a(n) = n * A111089(n). URL: https://oeis.org/A129598.

Commentary.

The initial value is a(1)=2. At every other natural n, a(n) is n times its greatest prime factor.

Definition 1.4 (The A050399 sequence).

$$\forall n\in \mathrm{Nat},b\left(n\right)=\mathrm{if}\left(\exists m\in \mathrm{Nat},(0<m)\wedge (g\left(m\right)=n),\mathrm{sInf}\left(\{m\in \mathrm{Nat}\mid ((0<m)\wedge (g\left(m\right)=n))\}\right),0\right)$$

Formalization. D5/S0/Certificates/KarttunenGcdTotientPositionRefutation.b (✓ std3).

Citation. Antti Karttunen (2007). OEIS A129598, a(n) = n * A111089(n). URL: https://oeis.org/A129598.

Commentary.

When a positive preimage of n under g exists, b(n) is the least such natural number; Nat.find supplies that least element. If the existence condition fails, the definition returns zero.

Definition 1.5 (Membership in A089966).

$$\forall n\in \mathrm{Nat},(inA089966\left(n\right))\iff ((n=1)\vee ((0<n)\wedge (\mathrm{card}\left(\mathrm{primeFactors}\left(n\right)\right)=\mathrm{greatestPrimeFactor}\left(n\right)\mathrm{mod}\mathrm{minFac}\left(n\right))))$$

Formalization. D5/S0/Certificates/KarttunenGcdTotientPositionRefutation.inA089966 (✓ std3).

Citation. Antti Karttunen (2007). OEIS A129598, a(n) = n * A111089(n). URL: https://oeis.org/A129598.

Commentary.

Membership holds at one, or at a positive n when the number of distinct prime factors equals the greatest prime factor modulo the least prime factor.

Definition 1.6 (Karttunen’s differing-position characterization).

$$(claim)\iff (\forall n\in \mathrm{Nat},(1<n)\Rightarrow ((a\left(n\right)\ne b\left(n\right))\iff (inA089966\left(n\right))))$$

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

Citation. Antti Karttunen (2007). OEIS A129598, a(n) = n * A111089(n). URL: https://oeis.org/A129598.

Commentary.

For every natural n greater than one, the characterization identifies a(n) and b(n) as unequal exactly at the members of A089966.

Theorem 1.7 (The characterization fails at 60).

$$\neg claim$$

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

Resolves. Problems/oeis-a129598-gcd-totient-position-refutation (refuted) by D5/S0/Certificates/KarttunenGcdTotientPositionRefutation.result.

Source. Repository-derived.

Acknowledgement. Antti Karttunen (2007). OEIS A129598, a(n) = n * A111089(n). URL: https://oeis.org/A129598.

Commentary.

At n=60, a(60)=300 has gcd-totient value 20 rather than 60, while b(60) has gcd-totient value 60 because 900 supplies a positive preimage. Thus a(60) and b(60) differ, but 60 is not in A089966. This refutes only the differing-position characterization; Conjecture 2 is untouched.

References

• Truth anchor: D5/S0/Certificates/KarttunenGcdTotientPositionRefutation.a
• Truth anchor: D5/S0/Certificates/KarttunenGcdTotientPositionRefutation.b
• Truth anchor: D5/S0/Certificates/KarttunenGcdTotientPositionRefutation.claim
• Truth anchor: D5/S0/Certificates/KarttunenGcdTotientPositionRefutation.g
• Truth anchor: D5/S0/Certificates/KarttunenGcdTotientPositionRefutation.greatestPrimeFactor
• Truth anchor: D5/S0/Certificates/KarttunenGcdTotientPositionRefutation.inA089966
• Truth anchor: D5/S0/Certificates/KarttunenGcdTotientPositionRefutation.result