

A194699


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


1



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, 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, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16
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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 FolsomKentOno theorem said: for a prime L >= 5, the partition numbers are Ladically fractal. Moreover, the Hausdorff dimension is <= floor((L  1)/12)  floor((L^2  1)/(24*L)). Also, the FolsomKentOno 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

Table of n, a(n) for n=1..77.
S. Ahlgren and K. Ono, Addition and counting: the arithmetic of partitions
A. Folsom, Z. A. Kent and K. Ono, ladic properties of the partition function, preprint.
A. Folsom, Z. A. Kent and K. Ono, ladic properties of the partition function, Advances in Mathematics, 229 (2012), pages 15861609.
Ken Ono (with Jan Bruinier, Amanda Folsom and Zach Kent), Emory University, Adding and counting
Wikipedia, Ramanujan's congruences


FORMULA

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

Cf. A000040, A000041, A182719, A194698.
Sequence in context: A195177 A147583 A054895 * A262694 A137588 A033271
Adjacent sequences: A194696 A194697 A194698 * A194700 A194701 A194702


KEYWORD

nonn


AUTHOR

Omar E. Pol, Jan 18 2012


STATUS

approved



