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A370600
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Numbers m such that 4m + k is squarefree for k = 1..3.
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2
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0, 1, 3, 5, 7, 8, 9, 10, 14, 16, 17, 19, 21, 23, 25, 26, 27, 28, 32, 34, 35, 39, 41, 44, 45, 46, 48, 50, 52, 53, 54, 55, 57, 59, 63, 64, 66, 70, 71, 75, 77, 79, 80, 82, 86, 88, 89, 91, 95, 97, 98, 99, 100, 102, 104, 107, 108, 109, 111, 113, 115, 116, 117, 120
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OFFSET
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1,3
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COMMENTS
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The number p^2*m is never squarefree, hence, 4*m is likewise never squarefree. Since 2 is the smallest prime, we have at most 3 consecutive squarefree numbers.
The asymptotic density of this sequence is 4 * Product_{p prime} (1 - 3/p^2) = 4 * A206256 = 0.501947... . - Amiram Eldar, Apr 16 2024
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LINKS
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Michael De Vlieger, Plot f(m) at (x,y) = (m mod 361, -floor(m/361)), m = 0..130320, 4X exaggeration, where f(m) = A008966(4m + 1), A008966(4m + 2), A008966(4m + 3), the first term assigned red, second green, and third blue channel. Hence m in this sequence appear white, while those in A258332 appear black.
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FORMULA
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EXAMPLE
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For m = 0, all of {4(0)+1, 4(0)+2, 4(0)+3} = {1, 2, 3} are squarefree and composite; these are all squarefree semiprimes. Hence, 0 is in the sequence.
For m = 2, {4(2)+1, 4(2)+2, 4(2)+3} = {9, 10, 11} only the latter 2 numbers are squarefree. Therefore, 2 is not in the sequence.
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MATHEMATICA
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Reap[Do[If[AllTrue[4 n + {1, 2, 3}, SquareFreeQ], Sow[n]], {n, 0, 120}] ][[-1, 1]]
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PROG
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(PARI) is(m) = issquarefree(4*m+1) && issquarefree(4*m+2) && issquarefree(4*m+3); \\ Amiram Eldar, Apr 16 2024
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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