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 A194699 #46 Jan 05 2023 10:11:37 %S A194699 0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6, %T A194699 6,6,7,7,7,7,7,8,8,8,8,8,9,9,9,10,10,10,10,10,11,11,11,11,12,12,12,12, %U A194699 13,13,13,13,14,14,14,15,15,15,15,16,16,16,16 %N A194699 a(n) = floor((p - 1)/12) - floor((p^2 - 1)/(24*p)), where p = prime(n). %C A194699 Sequence related to Ramanujan's famous partition congruences modulo powers of 5, 7 and 11. Ramanujan wrote: "It appears there are no equally simple properties for any moduli involving primes other than these three". On the other hand the Folsom-Kent-Ono theorem said: for a prime L >= 5, the partition numbers are L-adically fractal. Moreover, the Hausdorff dimension is <= floor((L - 1)/12) - floor((L^2 - 1)/(24*L)). Also, the Folsom-Kent-Ono corollary said: the dim is 0 only for L = 5, 7, 11 and so we have: 1) Ramanujan's congruences powers of 5, 7 and 11. 2) There are no simple properties for any other primes. %H A194699 S. Ahlgren and K. Ono, <a href="https://web.archive.org/web/20190728125806/http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/058.pdf">Addition and counting: the arithmetic of partitions</a> %H A194699 A. Folsom, Z. A. Kent and K. Ono, <a href="http://www.aimath.org/news/partition/folsom-kent-ono.pdf">l-adic properties of the partition function</a>, preprint. %H A194699 A. Folsom, Z. A. Kent and K. Ono, <a href="https://doi.org/10.1016/j.aim.2011.11.013">l-adic properties of the partition function</a>, Advances in Mathematics, 229 (2012), pages 1586-1609. %H A194699 Ken Ono (with Jan Bruinier, Amanda Folsom and Zach Kent), Emory University, <a href="http://www.youtube.com/watch?v=aj4FozCSg8g">Adding and counting</a> %H A194699 Wikipedia, <a href="http://en.wikipedia.org/wiki/Ramanujan's_congruences">Ramanujan's congruences</a> %F A194699 a(n) = A194698(A000040(n)). %F A194699 a(n) ~ 0.125 n log n. [_Charles R Greathouse IV_, Jan 25 2012] %e A194699 For primes 5, 7, 11 the Hausdorff dimension = 0, so a(3)..a(5) = 0. %e A194699 For primes 13, 17, 19, 23, 29, 31 the Hausdorff dimension = 1, so a(6)..a(11) = 1. %Y A194699 Cf. A000040, A000041, A182719, A194698. %K A194699 nonn %O A194699 1,12 %A A194699 _Omar E. Pol_, Jan 18 2012