bibkey: schulte2022a347293 authors: Werner Schulte year: 2022 title: “OEIS A347293, shifted-product gcd triangle and row-sum convolution” doi: null url: https://oeis.org/A347293 claim: “%N Triangle read by rows: T(n, k) = Sum_{i=1..n} gcd(1 + (i-1) * (k-1),n) for 1 <= k <= n. %F Conjecture: Row sums equal Dirichlet convolution of A000290 and A127473.” strata_touched:

• D5/S3/ArithSums/SchulteShiftedProductGcdRowSumConvolution license: citation-only triage: anchor

OEIS A347293

The entry defines a triangle by summing gcd(1+(i-1)(k-1),n) over i for each 1 <= k <= n. Reindexing with x=i-1 and y=k-1 identifies the row sum with the double sum of gcd(1+xy,n) over 0 <= x,y < n.

Its Formula section conjectures that these row sums are the Dirichlet convolution of A000290 and A127473. Here A000290 is the square function and A127473 is the square of Euler’s totient, so the asserted value at a positive n is Sum_{d|n} d^2 * phi(n/d)^2.

Verified locator

• URL: https://oeis.org/A347293
• %N (verbatim): Triangle read by rows: T(n, k) = Sum_{i=1..n} gcd(1 + (i-1) * (k-1),n) for 1 <= k <= n.
• %F (verbatim): Conjecture: Row sums equal Dirichlet convolution of A000290 and A127473.
• %A (verbatim): Werner Schulte, Jan 23 2022