OFFSET
1,12
COMMENTS
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.
LINKS
S. Ahlgren and K. Ono, Addition and counting: the arithmetic of partitions
A. Folsom, Z. A. Kent and K. Ono, l-adic properties of the partition function, preprint.
A. Folsom, Z. A. Kent and K. Ono, l-adic properties of the partition function, Advances in Mathematics, 229 (2012), pages 1586-1609.
Ken Ono (with Jan Bruinier, Amanda Folsom and Zach Kent), Emory University, Adding and counting
Wikipedia, Ramanujan's congruences
FORMULA
a(n) ~ 0.125 n log n. [Charles R Greathouse IV, Jan 25 2012]
EXAMPLE
For primes 5, 7, 11 the Hausdorff dimension = 0, so a(3)..a(5) = 0.
For primes 13, 17, 19, 23, 29, 31 the Hausdorff dimension = 1, so a(6)..a(11) = 1.
CROSSREFS
KEYWORD
nonn
AUTHOR
Omar E. Pol, Jan 18 2012
STATUS
approved