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A086550 Smallest k such that tau(k) - tau(k-1) = n, where tau(k) = number of divisors of k, or 0 if no such number exists.

%I A086550 #16 Aug 20 2017 17:56:50
%S A086550 3,2,6,50,12,36,24,400,48,1850,60,144,120,1600,168,576,180,1296,240,
%T A086550 4356,630,2304,360,900,960,9216,1008,40000,720,20736,840,5184,1800,
%U A086550 46656,1260,36864,1680,7056,3024,986050,2880,3600,6480,82944,2520,193600,3360
%N A086550 Smallest k such that tau(k) - tau(k-1) = n, where tau(k) = number of divisors of k, or 0 if no such number exists.
%C A086550 Conjecture: No term is zero.
%C A086550 a(2k+1) is either a square or one more than a square. - _David Wasserman_, Mar 24 2005
%H A086550 Giovanni Resta, <a href="/A086550/b086550.txt">Table of n, a(n) for n = 0..1000</a> (first 500 terms from Donovan Johnson)
%e A086550 a(3) = 50 as tau(50) - tau(49) = 6 - 3 = 3.
%t A086550 With[{tau=Partition[DivisorSigma[0,Range[10^6]],2,1]},Flatten[ Table[ Position[ #[[2]]-#[[1]]&/@tau,n,1,1],{n,0,50}]]]+1 (* _Harvey P. Dale_, Aug 20 2017 *)
%o A086550 (PARI) /* finds first 100 terms */ nn=vector(100); nd1=1; for(k=2, 24285184, nd2=numdiv(k); d=nd2-nd1; if(d>0, if(d<=100, if(nn[d]==0, nn[d]=k))); nd1=nd2); for(n=1, 100, write("b086550.txt", n " " nn[n])) /* _Donovan Johnson_, Sep 25 2013 */
%Y A086550 Cf. A285457.
%K A086550 nonn
%O A086550 0,1
%A A086550 _Amarnath Murthy_, Aug 28 2003
%E A086550 Corrected and extended by _David Wasserman_, Mar 24 2005
%E A086550 Offset changed to 0, and a(0) added by _Giovanni Resta_, Apr 28 2017