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A133466
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Positive integers k for which there is exactly one integer i in {1,2,3,...,k-1} such that i*k is a square.
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8
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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
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1,1
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COMMENTS
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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
The asymptotic density of this sequence is 3/(2*Pi^2). - Amiram Eldar, Mar 08 2021
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LINKS
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FORMULA
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{a(n)} = {A225546(A007283(n)) : n >= 0}, where {a(n)} denotes the set of integers in the sequence.
(End)
Sum_{n>=1} 1/a(n)^s = zeta(s)/(4^s*zeta(2*s)), s>1. - Amiram Eldar, Sep 26 2023
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EXAMPLE
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4 is in the sequence because among the products 1*4,2*4,3*4 = 4,8,12 there is exactly one square.
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MATHEMATICA
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eoiQ[n_]:=Count[n*Range[n-1], _?(IntegerQ[Sqrt[#]]&)]==1; Select[Range[ 400], eoiQ] (* Harvey P. Dale, Mar 14 2015 *)
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PROG
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(Haskell)
a133466 n = a133466_list !! (n-1)
a133466_list = map (+ 1) $ elemIndices 1 a057918_list
(PARI) isok(n) = sum(k=1, n-1, issquare(k*n)) == 1; \\ Michel Marcus, Nov 29 2019
(Magma) [k:k in [1..350]|#[m:m in [1..k-1]| IsSquare(m*k)] eq 1]; // Marius A. Burtea, Dec 03 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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