Yanev’s Sigma-Radical Identity

Abstract

Yanev’s identity expresses every positive divisor-power sum through ordinary divisor sums and the radical.

All variables take values in the natural numbers N. The named operator primeRadical is the frozen primeRadical of D5/S1/Deficit/AlmostAdditivity (A007947), the product of the distinct prime divisors, with empty product one. The notation sigma(k,x) denotes the sum of the k-th powers of the positive divisors of x, and is zero at x = 0. Powers, products and inequalities are in N, and m - 1 is truncated natural subtraction.

Theorem 1.1 (The general divisor-power identity).

$$\forall n:\mathbb{N},\forall m:\mathbb{N},(0<n)⟹(0<m)⟹\mathrm{sigma}\left(m,n\right)\cdot \mathrm{sigma}\left(1,\mathrm{primeRadical}{\left(n\right)}^{(m-1)}\right)=\mathrm{sigma}\left(1,n^m\cdot \mathrm{primeRadical}{\left(n\right)}^{(m-1)}\right)$$

Proof. Machine-checked in Lean as D5/S3/Arith/YanevSigmaRadicalIdentity.result (✓ std3). ∎

Resolves. Problems/oeis-a023887-yanev-sigma-radical-identity (proved) by D5/S3/Arith/YanevSigmaRadicalIdentity.result.

Citation. Olivier Gérard; Velin Yanev (2017). OEIS A023887, sigma_n(n), with Yanev’s sigma_m identity. URL: https://oeis.org/A023887.

Commentary.

For every n > 0 and m > 0, the displayed equation is the multiplied-out form of Yanev’s conjecture in A023887, using the frozen primeRadical of D5/S1/Deficit/AlmostAdditivity (A007947), rendered as the named operator primeRadical. Both sides are positive: the radical is positive and each divisor sum includes the divisor one. In particular sigma(1,primeRadical(n)^(m-1)) is positive, so division recovers the stated quotient, with exact natural-number division as well. Sela Fried (2025, Theorem 3) proved the m = 2 case on A001157; the general-m statement is the claim settled here. For prime powers, the equation follows from geometric-sum multiplication identities. Coprime multiplicativity of the radical and divisor sums then extends it to every positive natural number.

References

• Truth anchor: D5/S3/Arith/YanevSigmaRadicalIdentity.result
• Dependency: D5/S1/Deficit/AlmostAdditivity