A220506 -id:A220506 - cp's OEIS frontend

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

Omar E. Pol, Feb 04 2013

nonn

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

			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
...

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

Partial sums give A220506.

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

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

A222030 Primes and quarter-squares.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 11, 12, 13, 16, 17, 19, 20, 23, 25, 29, 30, 31, 36, 37, 41, 42, 43, 47, 49, 53, 56, 59, 61, 64, 67, 71, 72, 73, 79, 81, 83, 89, 90, 97, 100, 101, 103, 107, 109, 110, 113, 121, 127, 131, 132, 137, 139, 144, 149, 151, 156, 157, 163, 167, 169

Offset: 0

Omar E. Pol, Feb 05 2013

Union of A002620 and A000040.
It appears that there is always a prime between two consecutive quarter squares, if n >= 2. Therefore in a square spiral, or zig-zag, whose vertices are the quarter-squares, it appears that there is always a prime between two consecutive vertices, if n >= 2.
Apparently the above comment is equivalent to the Oppermann's conjecture. - Omar E. Pol, Oct 26 2013
Union of A000040 and A000290 and A002378. - Omar E. Pol, Oct 28 2013

Wikipedia, Oppermann's conjecture

Cf. A000040, A002620, A000290, A014085, A220492, A220506, A220508, A220516.

mx = 13; Union[Prime[Range[PrimePi[mx^2]]], Floor[Range[2*mx]^2/4]] (* Alonso del Arte, Mar 03 2013 *)

a(n) ~ n log n. - Charles R Greathouse IV, Mar 04 2013

cp's OEIS Frontend

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

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