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A080256
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Sum of numbers of distinct and of all prime factors of n.
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8
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0, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 5, 2, 4, 4, 5, 2, 5, 2, 5, 4, 4, 2, 6, 3, 4, 4, 5, 2, 6, 2, 6, 4, 4, 4, 6, 2, 4, 4, 6, 2, 6, 2, 5, 5, 4, 2, 7, 3, 5, 4, 5, 2, 6, 4, 6, 4, 4, 2, 7, 2, 4, 5, 7, 4, 6, 2, 5, 4, 6, 2, 7, 2, 4, 5, 5, 4, 6, 2, 7, 5, 4, 2, 7, 4, 4, 4, 6, 2, 7, 4, 5, 4, 4, 4, 8, 2, 5, 5, 6, 2, 6, 2, 6, 6
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OFFSET
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1,2
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COMMENTS
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a(n) = 2 iff n is prime, A000040; a(n) > 2 iff n is composite, A002808; a(n) <= 3 iff n is prime or square of prime, A000430; a(n) = 3 iff n is square of prime, A001248; a(A080257(n)) > 3;
a(n) <= 4 iff product of proper divisors <= n^2, A007964; a(n) = 4 iff n has four divisors, A030513; a(n) > 4 iff product of proper divisors > n^2, A058080; a(A064598(n)) <= 5; a(A080258(n)) = 5.
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LINKS
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FORMULA
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Additive with a(p^e) = e + 1.
Sum_{k=1..n} a(k) = 2 * n * log(log(n)) + c * n + O(n/log(n)), where c = A077761 + A083342 = 1.29615109474508069537... . - Amiram Eldar, Sep 28 2023
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MATHEMATICA
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f[n_] := Plus @@ (Last /@ FactorInteger[n] + 1); Table[ f[n], {n, 105}] (* Robert G. Wilson v, Aug 03 2005 *)
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PROG
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(PARI) a(n) = {my(f = factor(n)); omega(f) + bigomega(f); } \\ Amiram Eldar, Sep 28 2023
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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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