The OEIS A319927 Odd Non-Unitary Power-Sum Conjecture

Abstract

The sequence member 9216 refutes Ianakiev’s power-sum divisibility conjecture.

Definition 1.1 (Non-unitary divisors).

$$\forall n\in \mathrm{Nat},\mathrm{nonunitaryDivisors}\left(n\right)=\{d\in \mathrm{divisors}\left(n\right)\mid 1<\gcd \left(d,n/d\right)\}$$

Formalization. D5/S0/Certificates/IanakievOddNonunitaryPowerSumRefutation.nonunitaryDivisors (✓ std3).

Citation. Ivan N. Ianakiev (2018). OEIS A319927, Numbers k such that the sum of the squares of the odd non-unitary divisors of k divides the sum of the squares of the non-unitary divisors of k. URL: https://oeis.org/A319927.

Commentary.

The divisors d of n are filtered by gcd(d,n/d)>1. The slash denotes natural-number integer division; it is exact because every d in the displayed domain divides n.

Definition 1.2 (All non-unitary divisor powers).

$$\forall k\in \mathrm{Nat},n\in \mathrm{Nat},S\left(k,n\right)=\sum _{d\in \mathrm{nonunitaryDivisors}\left(n\right)}(d^k)$$

Formalization. D5/S0/Certificates/IanakievOddNonunitaryPowerSumRefutation.S (✓ std3).

Citation. Ivan N. Ianakiev (2018). OEIS A319927, Numbers k such that the sum of the squares of the odd non-unitary divisors of k divides the sum of the squares of the non-unitary divisors of k. URL: https://oeis.org/A319927.

Commentary.

For natural k and n, S(k,n) sums d^k over all non-unitary divisors d of n.

Definition 1.3 (Odd non-unitary divisor powers).

$$\forall k\in \mathrm{Nat},n\in \mathrm{Nat},O\left(k,n\right)=\sum _{d\in \{d\in \mathrm{nonunitaryDivisors}\left(n\right)\mid d\mathrm{mod}2=1\}}(d^k)$$

Formalization. D5/S0/Certificates/IanakievOddNonunitaryPowerSumRefutation.O (✓ std3).

Citation. Ivan N. Ianakiev (2018). OEIS A319927, Numbers k such that the sum of the squares of the odd non-unitary divisors of k divides the sum of the squares of the non-unitary divisors of k. URL: https://oeis.org/A319927.

Commentary.

For natural k and n, O(k,n) restricts the same sum to divisors with remainder one modulo two.

Definition 1.4 (Membership in A319927).

$$\forall n\in \mathrm{Nat},(\mathrm{member}\left(n\right))\iff ((0<n)\wedge \left((0<O\left(2,n\right))\wedge (O\left(2,n\right)\mid S\left(2,n\right))\right))$$

Formalization. D5/S0/Certificates/IanakievOddNonunitaryPowerSumRefutation.member (✓ std3).

Citation. Ivan N. Ianakiev (2018). OEIS A319927, Numbers k such that the sum of the squares of the odd non-unitary divisors of k divides the sum of the squares of the non-unitary divisors of k. URL: https://oeis.org/A319927.

Commentary.

A positive natural n is a member when O(2,n) is nonzero and divides S(2,n). The nonzero condition follows the PARI isok guard.

Definition 1.5 (Ianakiev’s conjecture).

$$(claim)\iff (\forall n\in \mathrm{Nat},(\mathrm{member}\left(n\right))\Rightarrow (\forall k\in \mathrm{Nat},O\left(k,n\right)\mid S\left(k,n\right)))$$

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

Citation. Ivan N. Ianakiev (2018). OEIS A319927, Numbers k such that the sum of the squares of the odd non-unitary divisors of k divides the sum of the squares of the non-unitary divisors of k. URL: https://oeis.org/A319927.

Commentary.

For every member n and every natural power k, the conjecture asserts that O(k,n) divides S(k,n).

Theorem 1.6 (The conjecture fails at 9216).

$$\neg claim$$

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

Resolves. Problems/oeis-a319927-odd-nonunitary-power-sum-refutation (refuted) by D5/S0/Certificates/IanakievOddNonunitaryPowerSumRefutation.result.

Source. Repository-derived.

Acknowledgement. Ivan N. Ianakiev (2018). OEIS A319927, Numbers k such that the sum of the squares of the odd non-unitary divisors of k divides the sum of the squares of the non-unitary divisors of k. URL: https://oeis.org/A319927.

Commentary.

At 9216=2^10*3^2 the odd non-unitary divisor set is {3}; O(1,9216)=3 and S(1,9216)=16361, so the claimed divisibility fails. Munn’s 2020 question about p=3 and the entry’s 2022 remark are not claimed here.

References

• Truth anchor: D5/S0/Certificates/IanakievOddNonunitaryPowerSumRefutation.O
• Truth anchor: D5/S0/Certificates/IanakievOddNonunitaryPowerSumRefutation.S
• Truth anchor: D5/S0/Certificates/IanakievOddNonunitaryPowerSumRefutation.claim
• Truth anchor: D5/S0/Certificates/IanakievOddNonunitaryPowerSumRefutation.member
• Truth anchor: D5/S0/Certificates/IanakievOddNonunitaryPowerSumRefutation.nonunitaryDivisors
• Truth anchor: D5/S0/Certificates/IanakievOddNonunitaryPowerSumRefutation.result