|
%I #20 Apr 21 2019 07:46:52
%S 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,
%T 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,
%U 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
%N a(n) = A000041(n) mod 10.
%C From _Johannes W. Meijer_, Jul 08 2011: (Start)
%C 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.
%C 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)
%D Robert Kanigel, The man who knew infinity: A life of the genius Ramanujan (1991) pp. 246-254 and pp. 299-307.
%H Seiichi Manyama, <a href="/A178743/b178743.txt">Table of n, a(n) for n = 0..10000</a>
%H Scott Ahlgren and Ken Ono, <a href="http://www.ams.org/notices/200109/fea-ahlgren.pdf">Addition and Counting: The Arithmetic of Partitions</a>, Notices of the AMS, 48 (2001) pp. 978-984.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PartitionFunctionPCongruences.html">Partition Function P Congruences</a>
%H <a href="/index/Fi#final">Index entries for sequences related to final digits of numbers</a>
%H <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>
%F a(n) = p(n) mod 10 with p(n) = A000041(n) the partition function.
%e From _Johannes W. Meijer_, Jul 08 2011: (Start)
%e d p(N=200) p(N=2000) p(N=4000) p(N=6000)
%e 0 0.16000 0.17750 0.17600 0.18067
%e 1 0.08500 0.08150 0.08125 0.07833
%e 2 0.08000 0.08400 0.08075 0.08033
%e 3 0.10000 0.08350 0.08150 0.07917
%e 4 0.05500 0.08050 0.07950 0.08233
%e 5 0.18500 0.16900 0.17625 0.17817
%e 6 0.08500 0.07500 0.07725 0.07867
%e 7 0.09000 0.08600 0.08700 0.08283
%e 8 0.06500 0.07650 0.07450 0.07517
%e 9 0.09500 0.08650 0.08600 0.08433
%e Total 1.00000 1.00000 1.00000 1.00000 (End)
%t Table[ Mod[ PartitionsP@n, 10], {n, 0, 111}]
%o (PARI) a(n) = numbpart(n) % 10; \\ _Michel Marcus_, Apr 21 2019
%Y Cf. A000041, A040051.
%Y Cf. A141053 (F(5*n+3) and Benford’s Law). - _Johannes W. Meijer_, Jul 08 2011
%K nonn,base
%O 0,3
%A _Robert G. Wilson v_, Jun 08 2010
%E Edited by _N. J. A. Sloane_, Jun 08 2010
|
