The Echo-Square Coprime sequence: a new integer sequence with index-dependent arithmetic and digit-sum constraints
We introduce the Echo-Square Coprime (ESC) sequence, a greedy smallest-unused-integer construction binding each term to its own index through three simultaneous constraints: coprimality with n, a triangular digit sum, and a square echo requiring a(n) to be a perfect square exactly when n is. The conjunction produces sharp emergent spikes at every square index, and the sequence is provably not a permutation of the positive integers.
1 · Definition
Starting from a(1) = 1, each term is the smallest positive integer not yet used that is coprime to its own index n, has a triangular base-10 digit sum, and is a perfect square if and only if n is a perfect square. No single constraint is unusual alone; their conjunction, applied index-wise rather than term-wise, is what distinguishes ESC from the Golomb, EKG, and Yellowstone constructions, where the rule relates a term only to its predecessors.
2 · Spikes at the square indices
The square echo holds exactly by construction, but its quantitative consequence is emergent: at every perfect-square index the sequence spikes sharply upward, because integers that are simultaneously square, coprime to n, and digit-sum-triangular are far sparser than the pool at ordinary indices. At non-square indices the trend is nearly linear, a(n)/n ≈ 3.87; at n = 100 the sequence jumps to 26,569 — about 266 times its index.
3 · Not a permutation
The digit-sum condition is index-independent, so any integer whose digit sum is not triangular is permanently ineligible: it can never appear, at any index. Of the first 300 positive integers exactly 74 have a triangular digit sum, and precisely those 74 appear among the first 200 terms — so, unlike the Golomb and EKG sequences, the ESC sequence is provably not a permutation of the positive integers, with 226 of the first 300 integers excluded for a checkable reason.
4 · Novelty and open problems
The term list 1, 3, 10, 361, 6, 19, 12, 15, 64, 21, 24, … matches no sequence in the OEIS or on the general web, verified by site-restricted search with its limits stated honestly. Open questions include a rigorous existence proof for all n (only computational evidence and a density heuristic are known), the asymptotics of the constant 3.87, and natural generalizations — a cube echo, pentagonal digit sums, or other bases — each yielding its own spike structure.