A220492 Number of primes p between quarter-squares, Q(n) < p <= Q(n+1), where Q(n) = A002620(n).

0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 1, 4, 1, 2, 2, 2, 3, 3, 2, 2, 2, 4, 2, 4, 3, 1, 4, 2, 4, 3, 3, 3, 4, 4, 3, 4, 3, 2, 4, 4, 5, 4, 4, 4, 3, 4, 4, 4, 5, 4, 4, 4, 4, 5, 5, 5, 4, 6, 4, 4, 5, 5, 5, 7, 2, 3, 6, 6, 6, 6, 5, 8, 4, 5, 6, 5, 4, 7

Offset: 0

Author

Omar E. Pol, Feb 04 2013

Keywords

• nonn

Comments

It appears that a(n) > 0, if n > 1.

Apparently the above comment is equivalent to the Oppermann's conjecture. - Omar E. Pol, Oct 26 2013

For n > 0, also the number of primes per quarter revolution of the Ulam Spiral. The conjecture implies that there is at least one prime in every turn after the first. - Ruud H.G. van Tol, Jan 30 2024

Examples

			When the nonnegative integers are written as an irregular triangle in which the right border gives the quarter-squares without repetitions, a(n) is the number of primes in the n-th row of triangle. See below (note that the prime numbers are in parenthesis):
---------------------------------------
Triangle                          a(n)
---------------------------------------
0;                                 0
1;                                 0
(2);                               1
(3),   4;                          1
(5),   6;                          1
(7),   8,   9;                     1
10,  (11), 12;                     1
(13), 14,  15,   16;               1
(17), 18, (19),  20;               2
21,   22, (23),  24,  25;          1
26,   27,  28,  (29), 30;          1
...
		

Links

• Ruud H.G. van Tol, Table of n, a(n) for n = 0..10000
• Wikipedia, Oppermann's conjecture

Crossrefs

Partial sums give A220506.

Cf. A000040, A002620, A001477, A014085, A066888, A073882, A222030.

Programs

PARI

a(n) = #primes([n^2/4, (n+1)^2/4]); \\ Ruud H.G. van Tol, Feb 01 2024