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.

3, 2, 6, 50, 12, 36, 24, 400, 48, 1850, 60, 144, 120, 1600, 168, 576, 180, 1296, 240, 4356, 630, 2304, 360, 900, 960, 9216, 1008, 40000, 720, 20736, 840, 5184, 1800, 46656, 1260, 36864, 1680, 7056, 3024, 986050, 2880, 3600, 6480, 82944, 2520, 193600, 3360

Offset: 0

Author

Amarnath Murthy, Aug 28 2003

Keywords

• nonn

Comments

Conjecture: No term is zero.

a(2k+1) is either a square or one more than a square. - David Wasserman, Mar 24 2005

Examples

			a(3) = 50 as tau(50) - tau(49) = 6 - 3 = 3.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 0..1000 (first 500 terms from Donovan Johnson)

Crossrefs

Cf. A285457.

Programs

Mathematica

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 *)

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 */

Extensions

Corrected and extended by David Wasserman, Mar 24 2005

Offset changed to 0, and a(0) added by Giovanni Resta, Apr 28 2017