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A328244
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Numbers whose second arithmetic derivative (A068346) is a squarefree number (A005117).
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10
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6, 9, 10, 14, 18, 21, 22, 25, 30, 34, 38, 42, 46, 50, 57, 58, 62, 65, 66, 69, 70, 77, 78, 82, 85, 86, 93, 94, 99, 105, 114, 118, 121, 122, 125, 126, 130, 133, 134, 138, 142, 145, 146, 150, 154, 161, 165, 166, 169, 170, 174, 177, 182, 185, 186, 198, 201, 202, 206, 207, 209, 213, 214, 217, 221, 222, 230, 231, 237, 238, 242, 246, 253, 254, 255
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OFFSET
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1,1
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COMMENTS
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Numbers whose first, second or third arithmetic is prime (A157037, A192192, A328239) are all included in this sequence, because: (1) taking arithmetic derivative of a prime gives 1, which is squarefree, (2) primes themselves are squarefree, and (3) only squarefree numbers may have arithmetic derivative that is a prime.
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LINKS
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EXAMPLE
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For n=6, its first arithmetic derivative is A003415(6) = 5, and its second derivative is A003415(5) = 1, and 1 is a squarefree number (in A005117), thus 6 is included in this sequence.
For n=9, A003415(9) = 6, A003415(6) = 5, and 5, like all prime numbers, is squarefree, thus 9 is included in this sequence.
For n=14, A003415(14) = 9, A003415(9) = 6 = 2*3, and as 6 is squarefree, 14 is included in this sequence.
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PROG
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(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA328244(n) = { my(u=A003415(A003415(n))); (u>0 && issquarefree(u)); };
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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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