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Number of factorizations of n into distinct numbers with an odd number of distinct prime factors.
2

%I #16 Nov 18 2019 02:20:25

%S 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,2,1,1,2,1,3,1,1,

%T 1,1,1,1,1,2,1,2,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,3,1,1,1,4,1,2,1,1,

%U 1,2,1,2,1,1,1,1,1,2,1,2,2,1,1,3,1,1,1,2,1,3,1,1,1,1,1,3,1,1,1,1,1,2,1,2,2

%N Number of factorizations of n into distinct numbers with an odd number of distinct prime factors.

%H Antti Karttunen, <a href="/A328856/b328856.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F Dirichlet g.f.: Product_{k>=1} (1 + A030230(k)^(-s)).

%F a(n) <= A045778(n). - _Antti Karttunen_, Oct 29 2019

%e a(32) = 3 because 32 = 4 * 8 = 2 * 16.

%o (PARI) seq(n)={my(v=vector(n, k, k==1)); for(k=2, n, if(omega(k)%2, my(m=logint(n, k), p=(1 + x + O(x*x^m)), w=vector(n)); for(i=0, m, w[k^i]=polcoef(p, i)); v=dirmul(v, w))); v} \\ _Andrew Howroyd_, Oct 29 2019, In older versions of PARI, use polcoeff instead of polcoef. - _Antti Karttunen_, Oct 29 2019

%o (PARI) A328856(n, k=n) = (((n<=k)&&((1==n)||(omega(n)%2))) + sumdiv(n, d, if(d > 1 && d <= k && d < n && (omega(d)%2), A328856(n/d, d-1)))); \\ _Antti Karttunen_, Oct 29 2019

%Y Cf. A030230, A045778, A050332, A328855.

%K nonn

%O 1,8

%A _Ilya Gutkovskiy_, Oct 28 2019

%E More terms from _Antti Karttunen_, Oct 29 2019