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A368460
a(n) = card(k: prime(n)^2 <= k < prime(n + 1)^2 and k term of A368458).
4
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, 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, 7, 22
OFFSET
1,5
COMMENTS
If A368458 is written as an irregular triangle for n >= 3, then a(n) is the length of row n.
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.
EXAMPLE
a(11) = 5 because 31^2 = 961, 1073, 1147, 1271, 1333, 1369 = 37^2 and all the terms are in that order in A368458.
PROG
(SageMath) # using A368207
def A368460(n):
pn = nth_prime(n); pn1 = nth_prime(n + 1)
A368457 = lambda n: 2 * (A368207(n) - sigma(n)) + n + 1
return sum(1 for n in range(pn ** 2, pn1 ** 2) if A368457(n) > 0)
print([A368460(n) for n in range(1, 25)])
CROSSREFS
Cf. A000203, A001248, A050216 (Brocard's Conjecture), A368207 (Bacher), A368457, A368458.
Sequence in context: A360058 A194332 A322062 * A350679 A071451 A177868
KEYWORD
nonn,more
AUTHOR
Peter Luschny, Dec 26 2023
STATUS
approved