A136404 Square numbers with more divisors than any smaller square number.

1, 4, 16, 36, 144, 576, 900, 3600, 14400, 32400, 44100, 129600, 176400, 705600, 1587600, 2822400, 6350400, 21344400, 57153600, 85377600, 192099600, 341510400, 768398400, 3073593600, 6915585600, 12294374400, 14428814400, 32464832400, 57715257600, 129859329600

Offset: 1

Author

J. Lowell, Mar 30 2008

Keywords

• nonn

Comments

Being the square of a number in A002182 is neither necessary nor sufficient.

Conjecture: square roots of the terms of this sequence are the same terms as A126098.

These are the squares of the indices of the RECORDS of A048691. - R. J. Mathar, Apr 04 2008

The square roots of terms are in A025487. - David A. Corneth, Oct 17 2018

From David A. Corneth, Oct 18 2018: (Start)

Records for largest exponents occur at: 1, 2, 5, 15, 25, 35, 200, 203

Least k such that a(k) divides prime(i)^4: 5, 10, 34, 104, 302

Based on these exponents I made the following dataset:

primorials <= 10^200 (92 such numbers).

Then made products of primorials <= 10^200 where the exponent of prime(11) is at most 2. Then searched records here. The b-file is checking A025487 squared checked. (End)

Examples

			900 qualifies because 576 has only 21 divisors and 900 has 27. 1296 does not because 1296 has only 25 divisors as opposed to the 27 of the smaller 900.
		

Links

• Ray Chandler, Table of n, a(n) for n = 1..582 (first 78 terms from Donovan Johnson, terms to 320 from David A. Corneth)
• David A. Corneth, Conjectured first 1350 terms

Crossrefs

Cf. A002182, A025487.

Programs

Maple

a := 0 : for n from 1 to 1000000 do ndvs := numtheory[tau](n^2) ; if ndvs > a then printf("%d,",n^2) ; a := ndvs ; fi ; od: # R. J. Mathar, Apr 04 2008
with(numtheory): a:=proc(n) if max(seq(tau(j^2),j=1..n-1))Emeric Deutsch, Apr 04 2008

Mathematica

With[{s = Array[DivisorSigma[0, #^2] &, 10^6]}, Map[FirstPosition[s, #][[1]]^2 &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Oct 15 2018 *)

Extensions

More terms from R. J. Mathar and Donovan Johnson, Apr 04 2008