login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Number of triples (u,v,w) of divisors of n with v-u = w-v, and u < v < w.
10

%I #13 Sep 11 2018 17:02:42

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

%C a(A091014(n))=n and a(m)<>n for m<=A091014(n);

%C a(A091010(n))=0; a(A091011(n))>0; a(A091012(n))=1; a(A091013(n))>1.

%C 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 Antti Karttunen, <a href="/A091009/b091009.txt">Table of n, a(n) for n = 1..65537</a>

%e 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 Array[Count[Subsets[#, {3}], _?(#2 - #1 == #3 - #2 & @@ # &)] &@ Divisors@ # &, 105] (* _Michael De Vlieger_, Sep 10 2018 *)

%o (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 Cf. also A094518.

%K nonn

%O 1,12

%A _Reinhard Zumkeller_, Dec 13 2003

%E Definition clarified by _Antti Karttunen_, Sep 10 2018