A141053 -id:A141053 - cp's OEIS frontend

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

Robert G. Wilson v, Jun 08 2010

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)

			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)

Robert Kanigel, The man who knew infinity: A life of the genius Ramanujan (1991) pp. 246-254 and pp. 299-307.

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

Cf. A000041, A040051.

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

Table[ Mod[ PartitionsP@n, 10], {n, 0, 111}]

a(n) = numbpart(n) % 10; \\ Michel Marcus, Apr 21 2019

a(n) = p(n) mod 10 with p(n) = A000041(n) the partition function.

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

cp's OEIS Frontend

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

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