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A304095 a(n) is the number of the proper divisors of n that are Lucas numbers larger than 3 (4, 7, 11, 18, ...).

%I A304095 #12 Jul 05 2025 11:25:07
%S A304095 0,0,0,0,0,0,0,1,0,0,0,1,0,1,0,1,0,0,0,1,1,1,0,1,0,0,0,2,0,0,0,1,1,0,
%T A304095 1,2,0,0,0,1,0,1,0,2,0,0,0,1,1,0,0,1,0,1,1,2,0,1,0,1,0,0,1,1,0,1,0,1,
%U A304095 0,1,0,2,0,0,0,1,2,0,0,1,0,0,0,2,0,0,1,2,0,1,1,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0,2,0,1,0,2,0,0,0,2,0,0,1,1
%N A304095 a(n) is the number of the proper divisors of n that are Lucas numbers larger than 3 (4, 7, 11, 18, ...).
%C A304095 a(n) is the number of the proper divisors d of n that are of the form d = A000045(k-1) + A000045(k+1), for k >= 3.
%H A304095 Antti Karttunen, <a href="/A304095/b304095.txt">Table of n, a(n) for n = 1..65537</a>
%F A304095 a(n) = Sum_{d|n, d>3, d<n} A102460(d).
%F A304095 a(n) = A007949(A304102(n)).
%F A304095 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A093540 - 4/3 = 0.629524... . - _Amiram Eldar_, Jul 05 2025
%e A304095 The proper divisors of 28 are 1, 2, 4, 7 and 14. Of these 4 and 7 are Lucas numbers (A000032) larger than 3, thus a(28) = 2.
%o A304095 (PARI)
%o A304095 A102460(n) = { my(u1=1,u2=3,old_u1); if(n<=2,sign(n),while(n>u2,old_u1=u1;u1=u2;u2=old_u1+u2);(u2==n)); };
%o A304095 A304095(n) = sumdiv(n,d,(d>3)*(d<n)*A102460(d));
%Y A304095 Cf. A000045, A000204, A093540, A102460, A304091, A304093, A304096, A304102.
%Y A304095 Cf. also A300836.
%K A304095 nonn
%O A304095 1,28
%A A304095 _Antti Karttunen_, May 13 2018