OFFSET
0,3
COMMENTS
From Johannes W. Meijer, Jul 08 2011: (Start)
We observe for the last digit a(n) of the partition function p(n) = A000041(n) that the probabilities of p(d = 0) = 0.18 and p(d = 5) = 0.18 while for the other digits p(d = 1, 2, 3, 4, 6, 7, 8, 9) = 0.08, see the examples. Ramanujan, who had access to the first two hundred p(n) thanks to MacMahon, observed this anomaly and subsequently proved that p(5*n+4) mod 5 = 0, see the references and links.
The first digit of the partition function p(n) follows Benford’s Law. This law states that the probability of having first digit d, 1 <= d <= 9, is p(d) = log_10(1+1/d), see the crossrefs. (End)
REFERENCES
Robert Kanigel, The man who knew infinity: A life of the genius Ramanujan (1991) pp. 246-254 and pp. 299-307.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000
Scott Ahlgren and Ken Ono, Addition and Counting: The Arithmetic of Partitions, Notices of the AMS, 48 (2001) pp. 978-984.
Eric Weisstein's World of Mathematics, Partition Function P Congruences
FORMULA
a(n) = p(n) mod 10 with p(n) = A000041(n) the partition function.
EXAMPLE
From Johannes W. Meijer, Jul 08 2011: (Start)
d p(N=200) p(N=2000) p(N=4000) p(N=6000)
0 0.16000 0.17750 0.17600 0.18067
1 0.08500 0.08150 0.08125 0.07833
2 0.08000 0.08400 0.08075 0.08033
3 0.10000 0.08350 0.08150 0.07917
4 0.05500 0.08050 0.07950 0.08233
5 0.18500 0.16900 0.17625 0.17817
6 0.08500 0.07500 0.07725 0.07867
7 0.09000 0.08600 0.08700 0.08283
8 0.06500 0.07650 0.07450 0.07517
9 0.09500 0.08650 0.08600 0.08433
Total 1.00000 1.00000 1.00000 1.00000 (End)
MATHEMATICA
Table[ Mod[ PartitionsP@n, 10], {n, 0, 111}]
PROG
(PARI) a(n) = numbpart(n) % 10; \\ Michel Marcus, Apr 21 2019
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Robert G. Wilson v, Jun 08 2010
EXTENSIONS
Edited by N. J. A. Sloane, Jun 08 2010
STATUS
approved