login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A071963(n) - n, where A071963(n) is the largest prime factor of p(n), the n-th partition number A000041(n).
4

%I #26 Jun 14 2023 15:16:23

%S 1,0,0,0,1,2,5,-2,3,-4,-3,-4,-1,88,-9,-4,-5,-6,-7,-12,-1,-10,145,228,

%T -17,64,3,16,-15,54,437,280,-9,-10,1197,6,17941,244,5,-28,87,152,2375,

%U 28,53,1042,195,20,6965,582,9233,610,1,5184,5,172,963,102302

%N A071963(n) - n, where A071963(n) is the largest prime factor of p(n), the n-th partition number A000041(n).

%C It appears that if n > 39, then a(n) is positive, i.e., A071963(n) > n. This has been checked up to n = 2500.

%C Cilleruelo and Luca proved that A071963(n) > log log n for almost all n, a much weaker statement. Earlier Schinzel and Wirsing proved that for all large N the product p(1)*p(2)*...*p(N) has at least C*log N distinct prime factors, for any positive constant C < 1/log 2.

%H T. D. Noe, <a href="/A192885/b192885.txt">Table of n, a(n) for n = 0..1000</a>

%H J. Cilleruelo and F. Luca, <a href="http://www.uam.es/personal_pdi/ciencias/cillerue/Papers/CLPofpofnAA.pdf">On the largest prime factor of the partition function of n</a>

%H A. Schinzel and E. Wirsing, <a href="http://dx.doi.org/10.1007/BF02837831">Multiplicative properties of the partition function</a>, Proc. Indian Acad. Sci., Math. Sci. (Ramanujan Birth Centenary Volume), 97 (1987), 297-303; <a href="https://www.ias.ac.in/describe/article/pmsc/097/01-03/0297-0303">alternative link</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GreatestPrimeFactor.html">Greatest Prime Factor</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PartitionFunctionP.html">Partition function</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Partition_(number_theory)">Partition function</a>

%F a(n) = A006530(A000041(n)) - n

%e There are 77 partitions of 12, and 77 = 7*11, so a(12) = 11 - 12 = -1.

%t Table[First[Last[FactorInteger[PartitionsP[n]]]] - n, {n, 0, 100}]

%o (PARI) a(n)=if(n<2,!n,my(f=factor(numbpart(n))[,1]);f[#f]-n) \\ _Charles R Greathouse IV_, Feb 04 2013

%Y Cf. A000041, A006530, A071963.

%K sign

%O 0,6

%A _Jonathan Sondow_, Aug 16 2011