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A086435 Maximum number of parts possible in a factorization of n into a product of distinct numbers > 1.

%I A086435 #16 Feb 16 2025 08:32:50
%S A086435 0,1,1,1,1,2,1,2,1,2,1,2,1,2,2,2,1,2,1,2,2,2,1,3,1,2,2,2,1,3,1,2,2,2,
%T A086435 2,3,1,2,2,3,1,3,1,2,2,2,1,3,1,2,2,2,1,3,2,3,2,2,1,3,1,2,2,3,2,3,1,2,
%U A086435 2,3,1,3,1,2,2,2,2,3,1,3,2,2,1,3,2,2,2,3,1,3,2,2,2,2,2,3,1,2,2,3,1,3,1,3,3
%N A086435 Maximum number of parts possible in a factorization of n into a product of distinct numbers > 1.
%C A086435 For n>1, a((n+1)!) = n is the first occurrence of n in the sequence. This function depends only on the prime signature of n. - _Franklin T. Adams-Watters_, Dec 19 2006
%C A086435 For integer n and prime p not dividing n, a(n*p) = a(n) + 1. - _Max Alekseyev_, Apr 23 2010
%H A086435 Antti Karttunen, <a href="/A086435/b086435.txt">Table of n, a(n) for n = 1..10000</a>
%H A086435 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnorderedFactorization.html">Unordered Factorization</a>.
%H A086435 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%e A086435 a(6)=2 since 6 may be factored into distinct parts as {{2,3},{6}}, so the largest number of factors possible is 2.
%e A086435 a(8)=2 since 8 may be factored into distinct parts as {{8},{2,4}}, so the largest numbers of factors possible is 2.
%o A086435 (PARI) { a(n,m=1) = if(n>m, 1 + vecmax( apply( x->if(x>m,a(n/x,x)), divisors(n) ))) } \\ _Max Alekseyev_, Jul 16 2009
%o A086435 (PARI) { aopt(n) = local(f,t); f=factor(n)[,2]; t=select(x->x>1,f); a(prod(j=1,#t,prime(j)^t[j])) + #f - #t } /* optimized version */ \\ _Max Alekseyev_, Apr 23 2010
%Y A086435 Cf. A000142, A025487.
%K A086435 nonn
%O A086435 1,6
%A A086435 _Eric W. Weisstein_, Jul 19 2003
%E A086435 More terms from _Max Alekseyev_, Apr 23 2010