login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A194699 a(n) = floor((p - 1)/12) - floor((p^2 - 1)/(24*p)), where p = prime(n). 1

%I

%S 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 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 13,13,13,13,14,14,14,15,15,15,15,16,16,16,16

%N a(n) = floor((p - 1)/12) - floor((p^2 - 1)/(24*p)), where p = prime(n).

%C 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 S. Ahlgren and K. Ono, <a href="http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/058.pdf">Addition and counting: the arithmetic of partitions</a>

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

%F a(n) = A194698(A000040(n)).

%F a(n) ~ 0.125 n log n. [_Charles R Greathouse IV_, Jan 25 2012]

%e For primes 5, 7, 11 the Hausdorff dimension = 0, so a(3)..a(5) = 0.

%e For primes 13, 17, 19, 23, 29, 31 the Hausdorff dimension = 1, so a(6)..a(11) = 1.

%Y Cf. A000040, A000041, A182719, A194698.

%K nonn

%O 1,12

%A _Omar E. Pol_, Jan 18 2012

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 28 03:57 EST 2020. Contains 331317 sequences. (Running on oeis4.)