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A091009 Number of triples (u,v,w) of divisors of n with v-u = w-v, and u < v < w.

%I A091009 #13 Sep 11 2018 17:02:42
%S A091009 0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,2,0,0,0,0,0,5,0,0,0,1,0,4,0,0,0,0,
%T A091009 0,6,0,0,0,1,0,2,0,0,3,0,0,7,0,0,0,0,0,3,0,2,0,0,0,11,0,0,0,0,0,3,0,0,
%U A091009 0,0,0,10,0,0,2,0,0,2,0,2,0,0,0,9,0,0,0,0,0,10,1,0,0,0,0,9,0,0,0,0,0
%N A091009 Number of triples (u,v,w) of divisors of n with v-u = w-v, and u < v < w.
%C A091009 a(A091014(n))=n and a(m)<>n for m<=A091014(n);
%C A091009 a(A091010(n))=0; a(A091011(n))>0; a(A091012(n))=1; a(A091013(n))>1.
%C A091009 Number of pairs (x,y) of divisors of n with x<y such that also 2y-x is a divisor of n. - _Antti Karttunen_, Sep 10 2018
%H A091009 Antti Karttunen, <a href="/A091009/b091009.txt">Table of n, a(n) for n = 1..65537</a>
%e A091009 a(30)=4, as there are exactly 4 triples of divisors with the defining property: (1,2,3), (1,3,5), (2,6,10) and (5,10,15).
%t A091009 Array[Count[Subsets[#, {3}], _?(#2 - #1 == #3 - #2 & @@ # &)] &@ Divisors@ # &, 105] (* _Michael De Vlieger_, Sep 10 2018 *)
%o A091009 (PARI) A091009(n) = if(1==n,0,my(d=divisors(n),c=0); for(i=1,(#d-1),for(j=(i+1),#d,if(!(n%(d[j]+(d[j]-d[i]))),c++))); (c)); \\ _Antti Karttunen_, Sep 10 2018
%Y A091009 Cf. also A094518.
%K A091009 nonn
%O A091009 1,12
%A A091009 _Reinhard Zumkeller_, Dec 13 2003
%E A091009 Definition clarified by _Antti Karttunen_, Sep 10 2018