

A133466


Positive integers k for which there is exactly one integer i in {1,2,3,...,k1} such that i*k is a square.


8



4, 8, 12, 20, 24, 28, 40, 44, 52, 56, 60, 68, 76, 84, 88, 92, 104, 116, 120, 124, 132, 136, 140, 148, 152, 156, 164, 168, 172, 184, 188, 204, 212, 220, 228, 232, 236, 244, 248, 260, 264, 268, 276, 280, 284, 292, 296, 308, 312, 316, 328, 332, 340, 344, 348, 356
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OFFSET

1,1


COMMENTS

It appears that all terms of this sequence are exactly four times those of the squarefree integers (A005117).
The observed behavior is true for all n. All positive integers n are written uniquely as k*m^2 where k is squarefree, k >=1, m >= 1. The square multiples of n are j^2*k*n, j >= 1. We seek n with exactly 1 multiple that is square and less than n^2. If m = 1, there are no such multiples as we have k = n, so the least square multiple is n^2. If m >= 2, k*n is square and less than n^2. However, 4*k*n also qualifies as square and less than n^2 if m > 2. So the qualifying values of n are those with m=2.  Peter Munn, Nov 28 2019
Numbers for which gcd(n,n')=4, where n' is the arithmetic derivative of n.  Paolo P. Lava Apr 24 2012
The asymptotic density of this sequence is 3/(2*Pi^2).  Amiram Eldar, Mar 08 2021


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000


FORMULA

A057918(a(n)) = 1.  Reinhard Zumkeller, Mar 27 2012
From Peter Munn, Nov 28 2019: (Start)
a(n) = 4 * A005117(n).
{a(n)} = {A225546(A007283(n)) : n >= 0}, where {a(n)} denotes the set of integers in the sequence.
(End)


EXAMPLE

4 is in the sequence because among the products 1*4,2*4,3*4 = 4,8,12 there is exactly one square.


MATHEMATICA

eoiQ[n_]:=Count[n*Range[n1], _?(IntegerQ[Sqrt[#]]&)]==1; Select[Range[ 400], eoiQ] (* Harvey P. Dale, Mar 14 2015 *)


PROG

(Haskell)
a133466 n = a133466_list !! (n1)
a133466_list = map (+ 1) $ elemIndices 1 a057918_list
 Reinhard Zumkeller, Mar 27 2012
(PARI) isok(n) = sum(k=1, n1, issquare(k*n)) == 1; \\ Michel Marcus, Nov 29 2019
(MAGMA) [k:k in [1..350]#[m:m in [1..k1] IsSquare(m*k)] eq 1]; // Marius A. Burtea, Dec 03 2019


CROSSREFS

Cf. A005117, A007283, A225546.
Cf. A057918, A195085.
Sequence in context: A311653 A171949 A217319 * A311654 A311655 A311656
Adjacent sequences: A133463 A133464 A133465 * A133467 A133468 A133469


KEYWORD

nonn


AUTHOR

John W. Layman, Nov 28 2007


STATUS

approved



