A194259 - OEIS

%I #40 Jun 14 2023 15:15:25

%S 0,1,2,3,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,8,9,9,10,11,12,13,14,15,16,

%T 17,17,18,19,20,21,21,21,22,23,24,25,27,28,30,31,32,33,34,35,36,37,39,

%U 40,42,43,44,45,47,48,49,51,52,53,54,56,57,59,60,61

%N Number of distinct prime factors of p(1)*p(2)*...*p(n), where p(n) is the n-th partition number.

%C Schinzel and Wirsing proved that a(n) > C*log n, for any positive constant C < 1/log 2 and all large n. In fact, it appears that a(n) > n for all n > 115 (see A194260).

%C It also appears that a(n) > a(n-1), for all n > 97, so that some prime factor of p(n) does not divide p(1)*p(2)*...*p(n-1). See A194261, A194262.

%H Alois P. Heinz and Giovanni Resta, <a href="/A194259/b194259.txt">Table of n, a(n) for n = 1..10000</a> (first 2000 terms from Alois P. Heinz)

%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/PartitionFunctionP.html">Partition Function</a>

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

%F a(n) = A001221(product(k=1..n, A000041(k))).

%e p(1)*p(2)*...*p(8) = 1*2*3*5*7*11*15*22 = 2^2 * 3^2 * 5^2 * 7 * 11^2, so a(8) = 5.

%p with(combinat): with(numtheory):

%p b:= proc(n) option remember;

%p       `if`(n=1, {}, b(n-1) union factorset(numbpart(n)))

%p     end:

%p a:= n-> nops(b(n)):

%p seq(a(n), n=1..100); # _Alois P. Heinz_, Aug 20 2011

%t a[n_] := Product[PartitionsP[k], {k, 1, n}] // PrimeNu; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Jan 28 2014 *)

%o (PARI) a(n)=my(v=[]);for(k=2,n,v=concat(v,factor(numbpart(k))[,1])); #vecsort(v,,8) \\ _Charles R Greathouse IV_, Feb 01 2013

%Y Cf. A000041, A001221, A087175, A194260, A194261, A194262.

%K nonn

%O 1,3

%A _Jonathan Sondow_, Aug 20 2011