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A178743 a(n) = A000041(n) mod 10. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified July 25 09:49 EDT 2021. Contains 346289 sequences. (Running on oeis4.)