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Number of primes p between quarter-squares, Q(n) < p <= Q(n+1), where Q(n) = A002620(n).
5

%I #52 Feb 03 2024 10:12:31

%S 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,

%T 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,

%U 4,5,5,5,7,2,3,6,6,6,6,5,8,4,5,6,5,4,7

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

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

%C Apparently the above comment is equivalent to the Oppermann's conjecture. - _Omar E. Pol_, Oct 26 2013

%C 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

%H Ruud H.G. van Tol, <a href="/A220492/b220492.txt">Table of n, a(n) for n = 0..10000</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Oppermann&#39;s_conjecture">Oppermann's conjecture</a>

%e 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):

%e ---------------------------------------

%e Triangle a(n)

%e ---------------------------------------

%e 0; 0

%e 1; 0

%e (2); 1

%e (3), 4; 1

%e (5), 6; 1

%e (7), 8, 9; 1

%e 10, (11), 12; 1

%e (13), 14, 15, 16; 1

%e (17), 18, (19), 20; 2

%e 21, 22, (23), 24, 25; 1

%e 26, 27, 28, (29), 30; 1

%e ...

%o (PARI) a(n) = #primes([n^2/4, (n+1)^2/4]); \\ _Ruud H.G. van Tol_, Feb 01 2024

%Y Partial sums give A220506.

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

%K nonn

%O 0,9

%A _Omar E. Pol_, Feb 04 2013