cp's OEIS Frontend

A368460 a(n) = card(k: prime(n)^2 <= k < prime(n + 1)^2 and k term of A368458).

%I A368460 #14 Dec 28 2023 02:05:30
%S A368460 1,1,1,1,2,2,2,3,3,2,5,3,3,4,5,4,3,9,5,4,7,5,6,14,6,4,7,3,5,22,6,6,4,
%T A368460 16,5,12,12,8,15,13,5,19,5,10,4,30,26,10,6,12,11,5,28,19,15,20,8,19,9,
%U A368460 7,22
%N A368460 a(n) = card(k: prime(n)^2 <= k < prime(n + 1)^2 and k term of A368458).
%C A368460 If A368458 is written as an irregular triangle for n >= 3, then a(n) is the length of row n.
%C A368460 Conjecture: For all n >= 5, there is at least one j such that b(j) = 2 * (Bacher(j) - sigma(j)) + j + 1 is > 0 and prime(n)^2 < b(j) < prime(n + 1)^2. In other words, a(n) > 1 for n >= 5.
%e A368460 a(11) = 5 because 31^2 = 961, 1073, 1147, 1271, 1333, 1369 = 37^2 and all the terms are in that order in A368458.
%o A368460 (SageMath) # using A368207
%o A368460 def A368460(n):
%o A368460     pn = nth_prime(n); pn1 = nth_prime(n + 1)
%o A368460     A368457 = lambda n: 2 * (A368207(n) - sigma(n)) + n + 1
%o A368460     return sum(1 for n in range(pn ** 2, pn1 ** 2) if A368457(n) > 0)
%o A368460 print([A368460(n) for n in range(1, 25)])
%Y A368460 Cf. A000203, A001248, A050216 (Brocard's Conjecture), A368207 (Bacher), A368457, A368458.
%K A368460 nonn,more
%O A368460 1,5
%A A368460 _Peter Luschny_, Dec 26 2023