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A275699
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Excess of numbers that are not squarefree.
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7
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1, 2, 1, 1, 3, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 1, 3, 1, 1, 3, 3, 1, 2, 1, 1, 4, 1, 1, 2, 2, 3, 3, 1, 1, 2, 1, 1, 2, 1, 6, 1, 2, 2, 1, 4, 1, 1, 1, 2, 1, 1, 4, 3, 1, 2, 1, 1, 1, 1, 3, 2, 2, 1, 2, 5, 2, 1, 3, 1, 1, 3, 1, 4, 1, 4, 2, 1
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OFFSET
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1,2
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COMMENTS
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The "excess" of a number is the number of prime divisors with multiplicity (the Omega function, A001222) minus the number of distinct prime divisors (the omega function, A001221). A046660(n) gives the excess of n.
Since squarefree numbers have no excess, this sequence is essentially A046660 with the 0's removed.
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LINKS
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FORMULA
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Asymptotic mean: lim_{m->oo} (1/m) Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p*(p-1)) / (1-6/Pi^2) = A136141/A229099 = 1.9719717... - Amiram Eldar, Feb 10 2021
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EXAMPLE
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Since 16 = 2^4, 16 has four prime divisors, but only one distinct divisor. Hence Omega(16) - omega(16) = 4 - 1 = 3. As 16 is the fifth number that is not squarefree, its corresponding 3 is a(5) in this sequence.
17 is prime and thus has no excess and no corresponding term in this sequence.
18 = 2 * 3^2, Omega(18) - omega(18) = 3 - 2 = 1, thus a(6) = 1.
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MATHEMATICA
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DeleteCases[Table[PrimeOmega[n] - PrimeNu[n], {n, 200}], 0] (* Alonso del Arte, Aug 05 2016 *)
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PROG
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(PARI) for(n=1, 200, if(bigomega(n)!=omega(n), print1(bigomega(n)-omega(n), ", ")))
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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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