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A194260 A194259(n) - n, where A194259(n) is the number of distinct prime factors of p(1)*p(2)*...*p(n) and p(n) is the n-th partition number.

%I A194260 #28 Jun 14 2023 15:16:02
%S A194260 -1,-1,-1,-1,-1,-1,-2,-3,-4,-5,-6,-7,-7,-8,-9,-10,-11,-12,-13,-13,-14,
%T A194260 -14,-14,-15,-15,-15,-15,-15,-15,-15,-15,-15,-16,-16,-16,-16,-16,-17,
%U A194260 -18,-18,-18,-18,-18,-17,-17,-16,-16,-16,-16,-16,-16,-16,-16,-15,-15,-14,-14,-14,-14,-13,-13,-13,-12,-12,-12,-12,-11,-11,-10,-10,-10,-10,-9,-9,-9,-9,-9,-8,-7,-7,-7,-8,-8,-8,-8,-7,-7,-7,-7,-6,-5,-4,-4,-4,-3,-3,-4,-4,-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,0,1
%N A194260 A194259(n) - n, where A194259(n) is the number of distinct prime factors of p(1)*p(2)*...*p(n) and p(n) is the n-th partition number.
%C A194260 Schinzel and Wirsing proved that a(n) > C*log n - n, for any positive constant C < 1/log 2 and all large n. In fact, it appears that a(n) > 0 for all n > 115.
%C A194260 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).
%H A194260 Alois P. Heinz and Giovanni Resta, <a href="/A194260/b194260.txt">Table of n, a(n) for n = 1..10000</a> (first 2000 terms from Alois P. Heinz)
%H A194260 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>.
%F A194260 a(n) = A001221(product(k=1..n, A000041(k))) - n.
%e A194260 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 - 8 = -3.
%p A194260 with(combinat): with(numtheory):
%p A194260 b:= proc(n) option remember;
%p A194260       `if`(n=1, {}, b(n-1) union factorset(numbpart(n)))
%p A194260     end:
%p A194260 a:= n-> nops(b(n)) -n:
%p A194260 seq(a(n), n=1..116); # _Alois P. Heinz_, Aug 20 2011
%t A194260 a[n_] := PrimeNu[Product[PartitionsP[k], {k, 1, n}]] - n; Table[a[n], {n, 1, 116}] (* _Jean-François Alcover_, Jan 28 2014 *)
%Y A194260 Cf. A000041, A001221, A087175, A194259.
%K A194260 sign
%O A194260 1,7
%A A194260 _Jonathan Sondow_, Aug 20 2011