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

0,1

Comments

Conjecture: No term is zero.

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

Links

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

Example

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

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 *)

Prog

(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 */

Crossrefs

Cf. A285457.

Sequence in context: A334588 A025240 A137602 * A266239 A018864 A271610

Adjacent sequences: A086547 A086548 A086549 * A086551 A086552 A086553

Keyword

nonn

Author

Amarnath Murthy, Aug 28 2003

Extensions

Corrected and extended by David Wasserman, Mar 24 2005

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

Status

approved