This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A220492 #52 Feb 03 2024 10:12:31 %S A220492 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 A220492 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 A220492 4,5,5,5,7,2,3,6,6,6,6,5,8,4,5,6,5,4,7 %N A220492 Number of primes p between quarter-squares, Q(n) < p <= Q(n+1), where Q(n) = A002620(n). %C A220492 It appears that a(n) > 0, if n > 1. %C A220492 Apparently the above comment is equivalent to the Oppermann's conjecture. - _Omar E. Pol_, Oct 26 2013 %C A220492 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 A220492 Ruud H.G. van Tol, <a href="/A220492/b220492.txt">Table of n, a(n) for n = 0..10000</a> %H A220492 Wikipedia, <a href="http://en.wikipedia.org/wiki/Oppermann's_conjecture">Oppermann's conjecture</a> %e A220492 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 A220492 --------------------------------------- %e A220492 Triangle a(n) %e A220492 --------------------------------------- %e A220492 0; 0 %e A220492 1; 0 %e A220492 (2); 1 %e A220492 (3), 4; 1 %e A220492 (5), 6; 1 %e A220492 (7), 8, 9; 1 %e A220492 10, (11), 12; 1 %e A220492 (13), 14, 15, 16; 1 %e A220492 (17), 18, (19), 20; 2 %e A220492 21, 22, (23), 24, 25; 1 %e A220492 26, 27, 28, (29), 30; 1 %e A220492 ... %o A220492 (PARI) a(n) = #primes([n^2/4, (n+1)^2/4]); \\ _Ruud H.G. van Tol_, Feb 01 2024 %Y A220492 Partial sums give A220506. %Y A220492 Cf. A000040, A002620, A001477, A014085, A066888, A073882, A222030. %K A220492 nonn %O A220492 0,9 %A A220492 _Omar E. Pol_, Feb 04 2013