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 A133466 Positive integers k for which there is exactly one integer i in {1,2,3,...,k-1} 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 (list; graph; refs; listen; history; text; internal format)
 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[n-1], _?(IntegerQ[Sqrt[#]]&)]==1; Select[Range[ 400], eoiQ] (* Harvey P. Dale, Mar 14 2015 *) PROG (Haskell) a133466 n = a133466_list !! (n-1) a133466_list = map (+ 1) \$ elemIndices 1 a057918_list -- Reinhard Zumkeller, Mar 27 2012 (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 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

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Last modified June 9 08:18 EDT 2023. Contains 363168 sequences. (Running on oeis4.)