A194699 a(n) = floor((p - 1)/12) - floor((p^2 - 1)/(24*p)), where p = prime(n).

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

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

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, 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) = 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