Integer sequences

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.