A166721 Squares for which no smaller square has the same number of divisors.

1, 4, 16, 36, 64, 144, 576, 900, 1024, 1296, 3600, 4096, 5184, 9216, 14400, 32400, 36864, 44100, 46656, 65536, 82944, 129600, 176400, 230400, 262144, 331776, 589824, 705600, 746496, 810000, 921600, 1166400, 1587600, 2073600, 2359296, 2822400, 2985984, 3240000

Offset: 1

Author

Alexander Isaev (i2357(AT)mail.ru), Oct 20 2009

Keywords

• easy
• nonn

Comments

From Jon E. Schoenfield, Mar 03 2018: (Start)

Numbers k^2 such there is no positive m < k such that A000005(m^2) = A000005(k^2).

Square terms in A007416. (End)

Examples

			The positive squares begin 1, 4, 9, 16, 25, 36, 49, 64, ..., and their corresponding numbers of divisors are 1, 3, 3, 5, 3, 9, 3, 7, ...; thus, a(1)=1, a(2)=4, 9 is not a term (it has the same number of divisors as does 4; the same is true of 25, 49, etc.), a(3)=16, a(4)=36, a(5)=64, ... - _Jon E. Schoenfield_, Mar 03 2018
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..300 from Alois P. Heinz)

Crossrefs

Cf. A000005, A005179, A007416, A025487, A048691, A136404, A166722.

Programs

Mathematica

 Sort[Module[{nn=2000,tbl},tbl=Table[{n^2,DivisorSigma[0,n^2]},{n,nn}];Table[ SelectFirst[ tbl,#[[2]]==k&],{k,nn}]][[All,1]]/."NotFound"->Nothing] (* Harvey P. Dale, Jun 06 2022 *)

PARI

lista(nn) = {v = []; for (n=1, nn, d = numdiv(n^2); if (! vecsearch(v, d), print1(n^2, ", "); v = Set(concat(v, d))););} \\ Michel Marcus, Mar 04 2018

Extensions

Proper definition and substantial editing by Jon E. Schoenfield, Mar 03 2018