A178743 a(n) = A000041(n) mod 10.

1, 1, 2, 3, 5, 7, 1, 5, 2, 0, 2, 6, 7, 1, 5, 6, 1, 7, 5, 0, 7, 2, 2, 5, 5, 8, 6, 0, 8, 5, 4, 2, 9, 3, 0, 3, 7, 7, 5, 5, 8, 3, 4, 1, 5, 4, 8, 4, 3, 5, 6, 3, 9, 1, 5, 6, 3, 4, 0, 0, 7, 5, 6, 9, 0, 8, 0, 9, 5, 5, 8, 5, 3, 9, 0, 4, 1, 3, 4, 0, 6, 7, 5, 9, 0, 7, 2, 3, 9, 5, 3, 9, 7, 7, 0, 9, 4, 0, 6, 5, 2, 6, 9, 0, 5

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

Index entries for sequences related to final digits of numbers

Index entries for sequences related to Benford's law

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

Cf. A000041, A040051.

Cf. A141053 (F(5*n+3) and Benford’s Law). - Johannes W. Meijer, Jul 08 2011

Sequence in context: A086355 A053666 A101987 * A126052 A321128 A130138

Adjacent sequences: A178740 A178741 A178742 * A178744 A178745 A178746

Keyword

nonn,base

Author

Robert G. Wilson v, Jun 08 2010

Extensions

Edited by N. J. A. Sloane, Jun 08 2010

Status

approved