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A349326
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a(n) is the number of prime powers (not including 1) that are bi-unitary divisors of n.
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2
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0, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 3, 2, 1, 3, 1, 5, 2, 2, 2, 2, 1, 2, 2, 4, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 2, 4, 2, 2, 1, 3, 1, 2, 2, 5, 2, 3, 1, 2, 2, 3, 1, 4, 1, 2, 2, 2, 2, 3, 1, 4, 3, 2, 1, 3, 2, 2, 2, 4, 1, 3, 2, 2, 2, 2, 2, 6, 1, 2, 2, 2, 1, 3, 1, 4, 3
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OFFSET
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1,6
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COMMENTS
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The total number of prime powers (not including 1) that divide n is A001222(n).
The least number k such that a(k) = m is A122756(m).
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LINKS
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FORMULA
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Additive with a(p^e) = e if e is odd, and e-1 if e is even.
a(n) <= A001222(n), with equality if and only if n is an exponentially odd number (A268335).
a(n) <= A286324(n) - 1, with equality if and only if n is a prime power (including 1, A000961).
a(n) = A001222(A350390(n)) (the number of primes factors of the largest exponentially odd number dividing n, counted with multiplicity).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B_2 - C), where B_2 = A083342 and C = A179119. (End)
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EXAMPLE
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12 has 4 bi-unitary divisors, 1, 3, 4 and 12. Two of these divisors, 3 and 4 = 2^2 are prime powers. Therefore a(12) = 2.
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MATHEMATICA
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f[p_, e_] := If[OddQ[e], e, e - 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
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PROG
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(PARI) a(n) = vecsum(apply(x -> if(x%2, x, x-1), factor(n)[, 2])); \\ Amiram Eldar, Sep 29 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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