cp's OEIS Frontend

a(n) is the index of last occurrence of n in A014085. This sequence relies on a heuristic calculation and there is no proof that it is correct. Conjecture: There is no k that has only one prime between k^2 and (k+1)^2.

Legendre's conjecture (still open) states that for n > 0 there is always a prime between n^2 and (n+1)^2. The number of primes between n^2 and (n+1)^2 is equal to A014085(n), so, the corresponding records are given by A014085(a(n)) = 0, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, ... (A349996).

m = 25 is the smallest number such that there are exactly 8 primes between m^2 = 625 et (m+1)^2 = 676, namely {631, 641, 643, 647, 653, 659, 661, 673} but there are 9 primes between 24^2 = 576 et 25^2 = 625, namely {577, 587, 593, 599, 601, 607, 613, 617, 619} so 24 is a term but not 25; hence, 25 is the first term of A076957 that is not a record.

This sequence is infinite. Suppose for contradiction that a(n) = k was the last term, with s primes between k^2 and (k+1)^2. Then there are at most s primes between (k+1)^2 and (k+2)^2, at most s primes between (k+2)^2 and (k+3)^3, and at most s*sqrt(x) + pi(k^2) primes up to x. But there are ~ x/log x primes up to x by the Prime Number Theorem, a contradiction. This can be made sharp with various explicit estimates. - Charles R Greathouse IV, Apr 10 2020